arXiv:1610.03625v2 [] 8 Apr 2017

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS
MARC KEILBERG
Abstract. We investigate a possible connection between the F SZ properties
of a group and its Sylow subgroups. We show that the simple groups G2 (5)
and S6 (5), as well as all sporadic simple groups with order divisible by 56 are
not F SZ, and that neither are their Sylow 5-subgroups. The groups G2 (5)
and HN were previously established as non-F SZ by Peter Schauenburg; we
present alternative proofs. All other sporadic simple groups and their Sylow
subgroups are shown to be F SZ. We conclude by considering all perfect groups
available through GAP with order at most 106 , and show they are non-F SZ
if and only if their Sylow 5-subgroups are non-F SZ.

Introduction
The F SZ properties for groups, as introduced by Iovanov et al. [4], arise from
considerations of certain invariants of the representation categories of semisimple
Hopf algebras known as higher Frobenius-Schur indicators [5, 10, 11]. See [9] for
a detailed discussion of the many important uses and generalizations of these invariants. When applied to Drinfeld doubles of finite groups, these invariants are
described entirely in group theoretical terms, and are in particular invariants of
the group itself. The F SZ property is then concerned with whether or not these
invariants are always integers—which gives the Z in F SZ.
While the F SZ and non-F SZ group properties are well-behaved with respect to
direct products [4, Example 4.5], there is currently little reason to suspect a particularly strong connection to proper subgroups which are not direct factors. Indeed,
by [2, 4] the symmetric groups Sn are F SZ, while there exist non-F SZ groups of
order 56 . Therefore, Sn is F SZ but contains non-F SZ subgroups for all sufficiently
large n. On the other hand, non-F SZ groups can have every proper subquotient
be F SZ. Even the known connection to the one element centralizers—see the comment following Definition 1.1—is relatively weak. In this paper we will establish a
few simple improvements to this situation, and then proceed to establish a number
of examples of F SZ and non-F SZ groups that support a potential connection to
Sylow subgroups. We propose this connection as Conjecture 2.7.
We will make extensive use of GAP [3] and the AtlasRep[15] package. Most of
the calculations were designed to be completed with only 2GB of memory or (much)
less available—in particular, using only a 32-bit implementation of GAP—, though
in a few cases a larger workspace was necessary. In all cases the calculations can
2010 Mathematics Subject Classification. Primary: 20D08; Secondary: 20F99, 16T05, 18D10.
Key words and phrases. sporadic groups, simple groups, Monster group, Baby Monster group,
Harada-Norton group, Lyons group, projective symplectic group, higher Frobenius-Schur indicators, FSZ groups, Sylow subgroups.
This work is in part an outgrowth of an extended e-mail discussion between Geoff Mason,
Susan Montgomery, Peter Schauenburg, Miodrag Iovanov, and the author. The author thanks
everyone involved for their contributions, feedback, and encouragement.
1

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MARC KEILBERG

be completed in workspaces with no more than 10GB of memory available. The
author ran the code on an Intel(R) Core(TM) i7-4770 CPU @ 3.40GHz machine
with 12GB of memory. All statements about runtime are made with respect to this
computer. Most of the calculations dealing with a particular group were completed
in a matter of minutes or less, though calculations that involve checking large
numbers of groups can take several days or more across multiple processors.
The structure of the paper is as follows. We introduce the relevant notation,
definitions, and background information in Section 1. In Section 2 we present a
few simple results which offer some connections between the F SZ (or non-F SZ)
property of G and certain of its subgroups. This motivates the principle investigation of the rest of the paper: comparing the F SZ properties for certain groups and
their Sylow subgroups. In Section 3 we introduce the core functions we will need to
perform our calculations in GAP. We also show that all groups of order less than
2016 (except possibly those of order 1024) are F SZ. The remainder of the paper
will be dedicated to exhibiting a number of examples that support Conjecture 2.7.
In Section 4 we show that the simple groups G2 (5), HN , Ly, B, and M , as
well as their Sylow 5-subgroups, are all non-F SZ5 . In Section 5 we show that all
other sporadic simple groups (including the Tits group) and their Sylow subgroups
are F SZ. This is summarized in Theorem 5.4. The case of the simple projective
symplectic group S6 (5) is handled in Section 6, which establishes S6 (5) as the second
smallest non-F SZ simple group after G2 (5). It follows from the investigations
of Schauenburg [13] that HN is then the third smallest non-F SZ simple group.
S6 (5) was not susceptible to the methods of Schauenburg [13], and requires further
modifications to our own methods to complete in reasonable time. We finish our
examples in Section 7 by examining those perfect groups available through GAP,
and show that they are F SZ if and only if their Sylow subgroups are F SZ. Indeed,
they are non-F SZ if and only if their Sylow 5-subgroup is non-F SZ5 .
Of necessity, these results also establish that various centralizers and maximal
subgroups in the groups in question are also non-F SZ5 , which can be taken as
additional examples. If the reader is interested in F SZ properties for other simple
groups, we note that Schauenburg [13] has checked all simple groups of order at
most |HN | = 273,030,912,000,000 = 214 ·36 ·56 ·7·11·19, except for S6 (5) (which we
resolve here); and that several families of simple groups were established as F SZ
by Iovanov et al. [4].
We caution the reader that the constant recurrence of the number 5 and Sylow
5-subgroups of order 56 in this paper is currently more of a computationally convenient coincidence than anything else. The reasons for this will be mentioned during
the course of the paper.
1. Background and Notation
Let N be the set of positive integers. The study of F SZ groups is connected to
the following sets.
Definition 1.1. Let G be a group, u, g ∈ G, and m ∈ N. Then we define
Gm (u, g) = {a ∈ G : am = (au−1 )m = g}.
Note that Gm (u, g) = ∅ if u 6∈ CG (g), and that in all cases Gm (u, g) ⊆ CG (g).
In particular, letting H = CG (g), then when u ∈ H we have
Gm (u, g) = Hm (u, g).

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

3

The following will then serve as our definition of the F SZm property. It’s equivalence to other definitions follows easily from [4, Corollary 3.2] and applications of
the Chinese remainder theorem.
Definition 1.2. A group G is F SZm if and only if for all g ∈ G, u ∈ CG (g), and
n ∈ N coprime to the order of g, we have
|Gm (u, g)| = |Gm (u, g n )|.
We say a group is F SZ if it is F SZm for all m.
The following result is useful for reducing the investigation of the F SZ properties
to the level of conjugacy classes or even rational classes.
Lemma 1.3. For any group G and u, g, x ∈ G we have a bijection Gm (u, g) →
Gm (ux , g x ) given by a 7→ ax .
If n ∈ N is coprime to |G| and r ∈ N is such that rn ≡ 1 mod |G|, we also have
a bijection Gm (u, g n ) → Gm (ur , g).
Proof. The first part is [5, Proposition 7.2] in slightly different notation. The second
part is [14, Corollary 5.5].

All expressions of the form Gm (u, g n ) will implicitly assume that n is coprime to
the order of g. We are free to replace n with an equivalent value which is coprime
to |G| whenever necessary. Moreover, when computing cardinalities |Gm (u, g)| it
suffices to compute the cardinalities |Hm (u, g)| for H = CG (g), instead. This latter
fact is very useful when attempting to work with groups of large order, or groups
with centralizers that are easy to compute in, especially when the group is suspected
of being non-F SZ.
+
, the union of which yields
Remark 1.4. There are stronger conditions called F SZm
+
the F SZ + condition, which are also introduced by Iovanov et al. [4]. The F SZm
condition is equivalent to the centralizer of every non-identity element with order
not in {1, 2, 3, 4, 6} being F SZm , which is in turn equivalent to the sets Gm (u, g) and
Gm (u, g n ) being isomorphic permutation modules for the two element centralizer
CG (u, g) [4, Theorem 3.8], with u, g, n satisfying the same constraints as for the
F SZm property. Here the action is by conjugation. We note that while the F SZ
property is concerned with certain invariants being in Z, the F SZ + property is not
concerned with these invariants being non-negative integers. When the invariants
are guaranteed to be non-negative is another area of research, and will also not be
considered here.

Example 1.5. The author has shown that quaternion groups and certain semidirect products defined from cyclic groups are always F SZ [7, 8]. This includes
the dihedral groups, semidihedral groups, and quasidihedral groups, among many
others.
Example 1.6. Iovanov et al. [4] showed that several groups and families of groups
are F SZ, including:
• All regular p-groups.
• Zp ≀r Zp , the Sylow p-subgroup of Sp2 , which is an irregular F SZ p-group.
• P SL2 (q) for a prime power q.

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MARC KEILBERG

• Any direct product of F SZ groups. Indeed, any direct product of F SZm
groups is also F SZm , as the cardinalities of the sets in Definition 1.1 split
over the direct product in an obvious fashion.
• The Mathieu groups M11 and M12 .
• Symmetric and alternating groups. See also [2].
Because of the first item, Susan Montgomery has proposed that we use the term
F S-regular instead of F SZ, and F S-irregular for non-F SZ. Similarly for F Sm regular and F Sm -irregular. These seem reasonable choices, but for this paper the
author will stick with the existing terminology.
Example 1.7. On the other hand, Iovanov et al. [4] also established that non-F SZ
groups exist by using GAP [3] to show that there are exactly 32 isomorphism classes
of groups of order 56 which are not F SZ5 .
Example 1.8. The author has constructed examples of non-F SZpj p-groups for
all primes p > 3 and j ∈ N in [6]. For j = 1 these groups have order pp+1 , which is
the minimum possible order for any non-F SZ p-group. Combined, [4, 6, 13] show,
among other things, that the minimum order of non-F SZ 2-groups is at least 210 ,
and the minimum order for non-F SZ 3-groups is at least 38 . It is unknown if any
non-F SZ 2-groups or 3-groups exist, however.
Example 1.9. Schauenburg [13] provides several equivalent formulations of the
F SZm properties, and uses them to construct GAP [3] functions which are useful
for testing the property. Using these functions, it was shown that the Chevalley
group G2 (5) and the sporadic simple group HN are not F SZ5 . These groups
were attacked directly, using advanced computing resources for HN , often with
an eye on computing the values of the indicators explicitly. We will later present
an alternative way of using GAP to prove that these groups, and their Sylow 5subgroups, are not F SZ5 . We will not attempt to compute the actual values of the
indicators, however.
One consequence of these examples is that the smallest known order for a nonF SZ group is 56 = 15,625. The groups with order divisible by pp+1 for p > 5
that are readily available through GAP are small in number, problematically large,
and frequently do not have convenient representations. Matrix groups have so far
proven too memory intensive for what we need to do, so we need permutation
or polycyclic presentations for accessible calculations. For these reasons, all of
the examples we pursue in the following sections will hone in on the non-F SZ5
property for groups with order divisible by 56 , and which admit known or reasonably
computable permutation representations. In most of the examples, 56 is the largest
power of 5 dividing the order, with the Monster group, the projective symplectic
group S6 (5), and the perfect groups of order 12 · 57 being the exceptions.
2. Obtaining the non-F SZ property from certain subgroups
Our first elementary result offers a starting point for investigating non-F SZm
groups of minimal order.
Lemma 2.1. Let G be a group with minimal order in the class of non-F SZm
groups. Then |Gm (u, g)| 6= |Gm (u, g n )| for some (n, |G|) = 1 implies g ∈ Z(G).
Proof. If not then CG (g) is a smaller non-F SZm group, a contradiction.



THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

5

The result applies to non-F SZm groups in a class that is suitably closed under
the taking of centralizers. For example, we have the following version for p-groups.
Corollary 2.2. Let P be a p-group with minimal order in the class of non-F SZpj
p-groups. Then |Ppj (u, g)| 6= |Ppj (u, g n )| for some p ∤ n implies g ∈ Z(P ).
Example 2.3. From the examples in the previous section, we know the minimum
possible order for a non-F SZp p-group for p > 3 is pp+1 . It remains unknown if
the examples of non-F SZpj p-groups from [6] for j > 1 have minimal order among
non-F SZpj p-groups. We also know that to check if a group of order 210 or 38 is
F SZ it suffices to assume that g is central.
Next, we determine a condition for when the non-F SZ property for a normal
subgroup implies the non-F SZ property for the full group.
Lemma 2.4. Let G be a group and suppose H is a non-F SZm normal subgroup
with m coprime to [G : H]. Then G is non-F SZm .
Proof. Let u, g ∈ H and (n, |g|) = 1 be such that |Hm (u, g)| 6= |Hm (u, g n )|. By
the index assumption, for all x ∈ G we have xm ∈ H ⇔ x ∈ H, so by definitions
Gm (u, g) = Hm (u, g) and Gm (u, g n ) = Hm (u, g n ), which gives the desired result.

Corollary 2.5. Let G be a finite group and suppose P is a normal non-F SZpj
Sylow p-subgroup of G for some prime p. Then G is non-F SZpj .
Corollary 2.6. Let G be a finite group and P a non-F SZpj Sylow p-subgroup of
G. Then the normalizer NG (P ) is non-F SZpj .
Sadly, we will find no actual use for Corollary 2.5 in the examples we consider
in this paper. However, this result, [13, Lemma 8.7], and the examples we collect
in the remainder of this paper do suggest the following conjectural relation for the
F SZ property.
Conjecture 2.7. A group is F SZ if and only if all of its Sylow subgroups are F SZ.
Some remarks on why this conjecture may involve some deep results to establish
affirmatively seems in order.
Consider a group G and let u, g ∈ G and n ∈ N with (n, |G|) = 1. Suppose that
g has order a power of p, for some prime p. Then
[
Gpj (u, g) =
Gpj (u, g) ∩ P x ,
where the union runs over all distinct conjugates P x in CG (g) of a fixed Sylow
p-subgroup P of CG (g). Let Ppxj (u, g) = Gpj (u, g) ∩ P x . Then |Gpj (u, g)| =
S
S
|Gpj (u, g n )| if and only if there is a bijection P x (u, g) → P x (u, g n ). In the
special case u ∈ P , if P was F SZpj we would have a bijection P (u, g) → P (u, g n ),
x
x
but this does not obviously guarantee a bijection
g n ) for all conS x P (u, g)
S →xP (u,
n
jugates. Attempting to get a bijection P (u, g) → P (u, g ) amounts, via
the Inclusion-Exclusion Principle, to controlling the intersections of any number of
conjugates and how many elements those intersections contribute to Gpj (u, g) and
Gpj (u, g n ). There is no easy or known way to predict the intersections of a collection of Sylow p-subgroups for a completely arbitrary G, so any positive affirmation
of the conjecture will impose a certain constraint on these intersections.

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MARC KEILBERG

Moreover, we have not considered the case of the sets Gm (u, g) where m has
more than one prime divisor, nor those cases where u, g do not order a power of
a fixed prime, so a positive affirmation of the conjecture is also expected to show
that the F SZm properties are all derived from the F SZpj properties for all prime
powers dividing m. On the other hand, a counterexample seems likely to involve
constructing a large group which exhibits a complex pattern of intersections in its
Sylow p-subgroups for some prime p, or otherwise exhibits the first example of a
group which is F SZpj for all prime powers but is nevertheless not F SZ.
Example 2.8. All currently known non-F SZ groups are either p-groups (for which
the conjecture is trivial), are nilpotent (so are just direct products of their Sylow
subgroups), or come from perfect groups (though the relevant centralizers need not
be perfect). The examples of both F SZ and non-F SZ groups we establish here
will also all come from perfect groups and p-groups. In the process we obtain,
via the centralizers and maximal subgroups considered, an example of a solvable,
non-nilpotent, non-F SZ group; as well as an example of a non-F SZ group which
is neither perfect nor solvable. All of these examples, of course, conform to the
conjecture.
3. GAP functions and groups of small order
The current gold standard for general purpose testing of the F SZ properties in
GAP [3] is the FSZtest function of Schauenburg [13]. In certain specific situations,
the function FSInd from [4] can also be useful for showing a group is non-F SZ.
However, with most of the groups we will consider in this paper both of these
functions are impractical to apply directly. The principle obstruction for FSZtest
is that this function needs to compute both conjugacy classes and character tables
of centralizers, and this can be a memory intensive if not wholly inaccessible task.
For FSInd the primary obstruction, beyond its specialized usage case, is that it
must completely enumerate, store, and sort the entire group (or centralizer). This,
too, can quickly run into issues with memory consumption.
We therefore need alternatives for testing (the failure of) the FSZ properties
which can sidestep such memory consumption issues. For Section 7 we will also
desire functions which can help us detect and eliminate the more ”obviously” FSZ
groups. We will further need to make various alterations to FSZtest to incorporate
these things, and to return a more useful value when the group is not F SZ.
The first function we need, FSZtestZ, is identical to FSZtest—and uses several of
the helper functions found in [13]—except that instead of calculating and iterating
over all rational classes of the group it iterates only over those of the center. It
needs only a single input, which is the group to be checked. If it finds that the
group is non-F SZ, rather than return false it returns the data that established
the non-F SZ property. Of particular importance are the values m and z. If the
group is not shown to be non-F SZ by this test, then it returns fail to indicate
that the test is typically inconclusive.
FSZtestZ := function (G)
l o c a l CT, zz , z , c l , div , d , c h i , m, b ;
c l := R a t i o n a l C l a s s e s ( Center (G) ) ;
c l := F i l t e r e d ( c l , c−>not Order ( R e p r e s e n t a t i v e ( c ) )

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

7

in [ 1 , 2 , 3 , 4 , 6 ] ) ;
for z z in c l do
z := R e p r e s e n t a t i v e ( z z ) ;
d i v := F i l t e r e d ( D i v i s o r s I n t ( Exponent (G) / Order ( z ) ) ,
m−>not Gcd (m, Order ( z ) ) in [ 1 , 2 , 3 , 4 , 6 ] ) ;
i f Length ( d i v ) < 1 then continue ; f i ;
CT := O r d i n a r y C h a r a c t er T a b le (G) ;
for c h i in I r r (CT) do
for m in d i v do
i f not I s R a t ( beta (CT, z , m, c h i ) )
then return [ z ,m, c h i ,CT ] ;
fi ;
od ;
od ;
od ;
#t h e t e s t i s i n c o n c l u s i v e i n g e n e r a l
return f a i l ;
end ;
This function is primarily useful for testing groups with minimal order in a class
closed under centralizers, such as in Lemma 2.1 and Corollary 2.2. Or for any group
with non-trivial center that is suspected of failing the F SZ property at a central
value.
We next desire a function which can quickly eliminate certain types of groups as
automatically being F SZ. For this, the following result on groups of small order is
helpful.
Theorem 3.1. Let G be a group with |G| < 2016 and |G| 6= 1024. Then G is F SZ.
Proof. By Lemma 2.1 it suffices to run FSZtestZ over all groups in the SmallGroups
library of GAP. This library includes all groups with |G| < 2016, except those of
order 210 = 1024. In practice, the author also used the function IMMtests introduced below, but where the check on the size of the group is constrained initially
to 100 by [4, Corollary 5.5], and can be increased whenever desired to eliminate all
groups of orders already completely tested. This boils down to quickly eliminating
p-groups and groups with relatively small exponent. By using the closure of the
F SZ properties with respect to direct products, one need only consider a certain
subset of the orders in question rather than every single one in turn, so as to avoid
essentially double-checking groups. We note that the groups of order 1536 take the
longest to check. The entire process takes several days over multiple processors,
but is otherwise straightforward.


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MARC KEILBERG

We now define the function IMMtests. This function implements most of the
more easily checked conditions found in [4] that guarantee the F SZ property, and
calls FSZtestZ when it encounters a suitable p-group. The function returns true
if the test conclusively establishes that the group is F SZ; the return value of
FSZtestZ if it conclusively determines the group is non-F SZ; and fail otherwise.
Note that whenever this function calls FSZtestZ that test is conclusive by Corollary 2.2, so it must adjust a return value of fail to true.
IMMtests := function (G)
l o c a l sz , b , l , p2 , p3 , po ;
i f I s A b e l i a n (G)
then return true ;
fi ;
s z := S i z e (G) ;
i f ( s z < 2 0 1 6 ) and ( not s z =1024)
then return true ;
fi ;
i f IsPGroup (G) then
#R e g u l a r p−g r o u p s are a l w a y s FSZ .
l := C o l l e c t e d ( F a c t o r s I n t ( s z ) ) [ 1 ] ;
i f l [1] >= l [ 2 ] or Exponent (G) = l [ 1 ]
then return true ;
fi ;
s z := Length ( U p p e r C e n t r a l S e r i e s (G) ) ;
i f l [ 1 ] = 2 then
i f l [2 ] < 1 0 or s z < 3
or Exponent (G)<64
then return true ;
e l i f l [ 2 ] = 1 0 and s z >= 3
then
b := FSZtestZ (G) ;
i f I s L i s t ( b ) then return b ;
e l s e return true ;
fi ;
fi ;
e l i f l [ 1 ] = 3 then
i f l [2 ] < 8 or s z < 4
or Exponent (G)<27

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

9

then return true ;
e l i f l [ 2 ] = 8 and sz >=4
then
b := FSZtestZ (G) ;
i f I s L i s t ( b ) then return b ;
e l s e return true ;
fi ;
fi ;
e l i f sz < l [1]+1
then return true ;
e l i f s z = l [ 1 ] + 1 and s z=l [ 2 ]
then
b := FSZtestZ (G) ;
i f I s L i s t ( b ) then return b ;
e l s e return true ;
fi ;
fi ;
else
#c h e c k t h e e x p o n e n t f o r non−p−g r o u p s
l := F a c t o r s I n t ( Exponent (G) ) ;
p2 := Length ( P o s i t i o n s ( l , 2 ) ) ;
p3 := Length ( P o s i t i o n s ( l , 3 ) ) ;
po := F i l t e r e d ( l , x−>x > 3 );
i f F o r A l l ( C o l l e c t e d ( po ) , x−>x [ 2 ] < 2 ) and
( ( p2 < 4 and p3 < 4 )
or ( p2 < 6 and p3 < 2 ) )
then return true ;
fi ;
fi ;
#t e s t s were i n c o n c l u s i v e
return f a i l ;
end ;
We then incorporate these changes into a modified version of FSZtest, which we
give the same name. Note that this function also uses the function beta and its
corresponding helper functions from [13]. It has the same inputs and outputs as
FSZtestZ, except that the test is definitive, and so returns true when the group is
F SZ.
FSZtest := function (G)
l o c a l C, CT, zz , z , c l , div , d , c h i , m, b ;
b := IMMtests (G ) ; ;

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MARC KEILBERG

i f not b= f a i l
then return b ;
fi ;
c l := R a t i o n a l C l a s s e s (G) ;
c l := F i l t e r e d ( c l , c−>not Order ( R e p r e s e n t a t i v e ( c ) )
in [ 1 , 2 , 3 , 4 , 6 ] ) ;
for z z in c l do
z := R e p r e s e n t a t i v e ( z z ) ;
C := C e n t r a l i z e r (G, z ) ;
d i v := F i l t e r e d ( D i v i s o r s I n t ( Exponent (C) / Order ( z ) ) ,
m−>not Gcd (m, Order ( z ) ) in [ 1 , 2 , 3 , 4 , 6 ] ) ;
i f Length ( d i v ) < 1 then continue ; f i ;
# Check f o r t h e e a s y c a s e s
b := IMMtests (C ) ;
i f b=true
then continue ;
e l i f I s L i s t ( b ) then
i f R a t i o n a l C l a s s (C, z)= R a t i o n a l C l a s s (C, b [ 2 ] )
then return b ;
fi ;
fi ;
CT := O r d i n a r y C h a r a c t er T a b le (C) ;
for c h i in I r r (CT) do
for m in d i v do
i f not I s R a t ( beta (CT, z , m, c h i ) )
then return [m, z , c h i ,CT ] ;
fi ;
od ;
od ;
od ;
return true ;
end ;
Our typical procedure will be as follows: given a group G, take its Sylow 5subgroup P and find u, g ∈ P such that |P5 (u, g)| 6= |P5 (u, g n )| for 5 ∤ n, and
then show that |G5 (u, g)| 6= |G5 (u, g n )|. The second entry in the list returned by

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

11

FSZtest gives precisely the g value we need. But it does not provide the u value
directly, nor the n. As it turns out, we can always take n = 2 when o(g) = 5, but
for other orders this need not necessarily hold.
In order to acquire these values we introduce the function FSIndPt below, which
is a variation on FSInd [4]. This function has the same essential limitation that
FSInd does, in that it needs to completely enumerate, store, and sort the elements
of the group. This could in principle be avoided, at the cost of increased run-time.
However our main use for the function is to apply it to Sylow 5-subgroups which
have small enough order that this issue does not pop up.
The inputs are a group G, m ∈ N and g ∈ G. It is best if one in fact passes in
CG (g) for G, but the function will compute the centralizer regardless. The function
looks for an element u ∈ CG (g) and an integer j coprime to the order of g such that
|Gm (u, g)| 6= |Gm (u, g j )|. The output is the two element list [u,j] if such data
exists, otherwise it returns fail to indicate that the test is normally inconclusive.
Note that by Lemma 1.3 and centrality of g in C = CG (g) we need only consider
the rational classes in C to find such a u.
FSIndPt:= function (G,m, g )
l o c a l GG, C, Cl , g u c o e f f , elG , Gm, a l i s t ,
a u l i s t , u m l i s t , npos , j , n , u , pr ;
C := C e n t r a l i z e r (G, g ) ;
GG := EnumeratorSorted (C ) ; ;
elG := S i z e (C ) ;
Gm := L i s t (GG, x−>P o s i t i o n (GG, xˆm) ) ;
pr := P r imeResidues ( Order ( g ) ) ;
for Cl in R a t i o n a l C l a s s e s (C) do
u := R e p r e s e n t a t i v e ( Cl ) ;
npos := [ ] ;
a l i s t := [ ] ;
a u l i s t := [ ] ;
u m l i s t := [ ] ;
g u c o e f f := [ ] ;
u m l i s t := L i s t (GG, a−>P o s i t i o n (GG,
( a∗ I n v e r s e ( u ) ) ˆm) ) ; ;
#The f o l l o w i n g computes t h e c a r d i n a l i t i e s
# o f G m( u , g ˆn ) .
for n in pr

12

MARC KEILBERG

do
npos := P o s i t i o n (GG, g ˆn ) ;
a l i s t := P o s i t i o n s (Gm, npos ) ;
a u l i s t := P o s i t i o n s ( u m l i s t , npos ) ;
g u c o e f f [ n ] := S i z e (
Intersection ( alist , aulist ));
#Check i f we ’ ve fou n d our u
i f not g u c o e f f [ n ] = g u c o e f f [ 1 ]
then return [ u , n ] ;
fi ;
od ;
od ;
#No u was fou n d f o r t h i s G,m, g
return f a i l ;
end ;
Lastly, we introduce the function FSZSetCards, which is the most naive and
straightforward way of computing both |Gm (u, g)| and |Gm (u, g n )|. The inputs are
a set C of group elements—normally this would be CG (g), but could be a conjugacy
class or some other subset or subgroup—; group elements u, g; and integers m, n
such that g 6= g n . The output is a two element list, which counts the number of
elements of C in Gm (u, g) in the first entry and the number of elements of C in
Gm (u, g n ) in the second entry. It is left to the user to check that the inputs satisfy
whatever relations are needed, and to then properly interpret the output.
FSZSetCards := function (C, u , g ,m, n )
l o c a l c o n t r i b s , apow , aupow , a ;
c o n t r i b s := [ 0 , 0 ] ;
for a in C do
apow := a ˆm;
aupow := ( a ∗ I n v e r s e ( u ) ) ˆm;
i f ( apow = g and aupow = g ) then
c o n t r i b s [ 1 ] := c o n t r i b s [ 1 ] + 1 ;
e l i f ( apow=g ˆn and aupow=gˆn ) then
c o n t r i b s [ 2 ] := c o n t r i b s [ 2 ] + 1 ;
fi ;
od ;
return ( c o n t r i b s ) ;
end ;

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

13

As long as C admits a reasonable iterator in GAP then this function can compute
these cardinalities with a very minimal consumption of memory. Any polycyclic or
permutation group satisfies this, as well as any conjugacy class therein. However,
for a matrix group GAP will attempt to convert to a permutation representation,
which is usually very costly.
The trade-off, as it often is, is in the speed of execution. For permutation groups
the run-time can be heavily impacted by the degree, such that it is almost always
worthwhile to apply SmallerDegreePermutationRepresentation whenever possible. If the reader wishes to use this function on some group that hasn’t been tested
before, the author would advise adding in code that would give you some ability
to gauge how far along the function is. By default there is nothing in the above
code, even if you interrupt the execution to check the local variables, to tell you if
the calculation is close to completion. Due to a variety of technical matters it is
difficult to precisely benchmark the function, but when checking a large group it
is advisable to acquire at least some sense of whether the calculation may require
substantial amounts of time.
Remark 3.2. Should the reader opt to run our code to see the results for themselves,
they may occasionally find that the outputs of FSZSetCards occur in the opposite
order we list here. This is due to certain isomorphisms and presentations for groups
calculated in GAP not always being guaranteed to be identical every single time you
run the code. As a result, the values for u or g may sometimes be a coprime power
(often the inverse) of what they are in other executions of the code. Nevertheless,
there are no issues with the function proving the non-F SZ property thanks to
Lemma 1.3, and there is sufficient predictability to make the order of the output
the only variation.
While very naive, FSZSetCards will suffice for most of our purposes, with all
uses of it completing in an hour or less. However, in Section 6 we will find an
example where the expected run-time for this function is measured in weeks, and
for which FSZtest requires immense amounts of memory—Schauenburg [13] says
that FSZtest for this group consumed 128 GB of memory without completing!
We therefore need a slightly less naive approach to achieve a more palatable runtime in this case. We leave this to Section 6, but note to the reader that the method
this section uses can also be applied to all of the other groups for which FSZSetCards
suffices. The reason we bother to introduce and use FSZSetCards is that the method
of Section 6 relies on being able to compute conjugacy classes, which can hit memory
consumption issues that FSZSetCards will not encounter. It is not our goal with
these functions to find the most efficient, general-purpose procedure. Instead we
seek to highlight some of the ways in which computationally problematic groups
may be rendered tractable by altering the approach one takes, and to show that the
non-F SZ property of these groups can be demonstrated in a (perhaps surprisingly)
short amount of time and with very little memory consumption.
4. The non-F SZ sporadic simple groups
The goal for this section is to show that the Chevalley group G2 (5), and all
sporadic simple groups with order divisible by 56 , as well as their Sylow 5-subgroups,
are non-F SZ5 . We begin with a discussion of the general idea for the approach.
Our first point of observation is that the only primes p such that pp+1 divides the
order of any of these groups have p ≤ 5. Indeed, a careful analysis of the non-F SZ

14

MARC KEILBERG

groups of order 56 found in [4] shows that several of them are non-split extensions
with a normal extra-special group of order 55 , which can be denoted in AtlasRep
notation as 51+4 .5. Consulting the known maximal subgroups for these groups we
can easily infer that the Sylow 5-subgroups of HN , G2 (5), B, and Ly have this same
form, and that the Monster has such a p-subgroup. Indeed, G2 (5) is a maximal
subgroup of HN , and B and Ly have maximal subgroups containing a copy of
HN , so these Sylow subgroups are all isomorphic. Furthermore, the Monster’s
Sylow 5-subgroup has the form 51+6 .52 , a non-split extension of the elementary
abelian group of order 25 by an extra special group of order 57 . Given this, we
suspect that these Sylow 5-subgroups are all non-F SZ5 , and that this will cause
the groups themselves to be non-F SZ5 .
We can then exploit the fact that non-trivial p-groups all have non-trivial centers
to obtain centralizers in the parent group that contain a Sylow 5-subgroup. In the
case of G = HN or G = G2 (5), we can quickly find u, g ∈ P , with P a Sylow
5-subgroup of G, such that |P5 (u, g)| 6= |P5 (u, g 2 )|, and show that for H = CG (g)
we have |H5 (u, g)| 6= |H5 (u, g 2 )|. Since necessarily |H5 (u, g)| = |G5 (u, g)| and
|H5 (u, g 2 )| = |G5 (u, g 2 )|, this will show that HN and G2 (5) are non-F SZ5 . Unfortunately, it turns out that P is not normal in H in either case, so the cardinalities of
these sets in H must be checked directly, rather than simply applying Corollary 2.5.
The remaining groups require a little more work, for various reasons.
In the case of the Monster, there is a unique non-identity conjugacy class yielding
a centralizer with order divisible by 59 . So we are free to pick any subgroup G
of M that contains a centralizer with this same order. Fortunately, not only is
such a (maximal) subgroup known, but Bray and Wilson [1] have also computed
a permutation representation for it. This is available in GAP via the AtlasRep
package. This makes all necessary calculations for the Monster accessible. The
Sylow 5-subgroup is fairly easily shown to be non-F SZ5 directly. However, the
centralizer we get in this way has large order, and its Sylow 5-subgroup is not
normal, making it impractical to work with on a personal computer. However,
further consultation of character tables shows that the Monster group has a unique
conjugacy class of an element of order 10 whose centralizer is divisible by 56 . So we
may again pick any convenient (maximal) subgroup with such a centralizer, and it
turns out the same maximal subgroup works. We construct the appropriate element
of order 10 by using suitable elements from Sylow subgroups of the larger centralizer,
and similarly to get the element u. Again it turns out that the Sylow 5-subgroup of
this smaller subgroup is not normal, so we must compute the set cardinalities over
the entire centralizer in question. However, this centralizer is about 1/8000-th the
size of the initial one, and we are subsequently able to calculate the appropriate
cardinalities in under an hour.
The Baby Monster can then be handled by using the fact that the Monster
contains the double cover of B as the centralizer of an involution to obtain the centralizer we need in B from a centralizer in M . The author thanks Robert Wilson
for reminding them of this fact. For the Lyons group, the idea is much the same
as for HN and G2 (5), with the additional complication that the AtlasRep package
does not currently contain any permutation representations for Ly. To resolve this,
we obtain a permutation representation for Ly, either computed directly in GAP

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

15

or downloaded [12]. This is then used to construct a suitable permutation representation of the maximal subgroup in question. Once this is done the calculations
proceed without difficulties.
These calculations all make extensive use of the functions given in Section 3.
4.1. Chevalley group G2 (5). We now show that G2 (5) and its Sylow 5-subgroups
are not F SZ5 . This was independently verified in [13]. Since G2 (5) is of relatively
small order, it can be attacked quickly and easily.
Theorem 4.1. The simple Chevalley group G2 (5) and its Sylow 5-subgroup are
non-F SZ5 .
Proof. The claims follow from running the following GAP code.
G := AtlasGroup ( ”G2 ( 5 ) ” ) ; ;
P := SylowSubgroup (G, 5 ) ; ;
# The f o l l o w i n g shows P i s n ot FSZ 5
g := FSZtestZ (P ) [ 2 ] ;
# Find u
u := FSIndPt (P , 5 , g ) [ 1 ] ; ;
C := C e n t r a l i z e r (G, g ) ; ;
#Check t h e c a r d i n a l i t i e s
FSZSetCards (C, u , g , 5 , 2 ) ;
The output is [0,625], so it follows that G and P are both non-F SZ5 as desired. 
We note that P is not normal in C, and indeed C is a perfect group of order
375,000 = 23 · 3 · 56 .
The call to FSZSetCards above runs in approximately 11 seconds, which is approximately the amount of time necessary to run FSZtest on G2 (5) directly. In
this case, the use of FSZSetCards is not particularly efficient, as the groups in
question are of reasonably small sizes and permutation degree. Nevertheless, this
demonstrates the basic method we will employ for all subsequent groups.
4.2. The Harada-Norton group. For the group HN the idea proceeds similarly
as for G2 (5).
Theorem 4.2. The Harada-Norton simple group HN and its Sylow 5-subgroup are
not F SZ5 .
Proof. To establish the claims it suffices to run the following GAP code.
G := AtlasGroup ( ”HN” ) ; ;
P := SylowSubgroup (G, 5 ) ; ;
# G, t h u s P, has v e r y l a r g e d e g r e e .
# P o l y c y c l i c g r o u p s are e a s i e r t o work w i t h .
i s o P := IsomorphismPcGroup (P ) ; ;
P := Image ( i s o P ) ; ;

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MARC KEILBERG

#Find u , g f o r P
g := FSZtestZ (P ) [ 2 ] ;
u := FSIndPt (P , 5 , g ) [ 1 ] ;
g := Image ( Inver seGener a lMa pping ( i s o P ) , g ) ; ;
u := Image ( Inver seGener a lMa pping ( i s o P ) , u ) ; ;
C := C e n t r a l i z e r (G, g ) ; ;
iso C := IsomorphismPcGroup (C ) ; ;
C := Image ( iso C ) ; ;
FSZSetCards (C, Image ( isoC , u ) , Image ( isoC , g ) , 5 , 2 ) ;
This code executes in approximately 42 minutes, with approximately 40 of that
spent finding P . The final output is [3125,0], so we conclude that both P and HN
are non-F SZ5 , as desired.

P is again not a normal subgroup of C, so we again must test the entire centralizer
rather than just P . We note that |C| = 25 56 = 500,000. Indeed, C is itself nonF SZ5 of necessity, and the fact that the call to IsomorphismPcGroup did not fail
means that C is solvable, and in particular not perfect and not a p-group.
4.3. The Monster group. We will now consider the Monster group M . The full
Monster group is famously difficult to compute in. But, as detailed in the beginning of the section, by consulting character tables of M and its known maximal
subgroups, we can find a maximal subgroup which contains a suitable centralizer
(indeed, two suitable centralizers) and also admits a known permutation representation [1].
Theorem 4.3. The Monster group M and its Sylow 5-subgroup are not F SZ5 .
Proof. The Sylow 5-subgroup of M has order 59 . Consulting the character table of
M , we see that M has a unique conjugacy class yielding a proper centralizer with
order divisible by 59 , and a unique conjugacy class of an element of order 10 whose
centralizer has order divisible by 56 ; moreover, the order of the latter centralizer is
precisely 12 million, and in particular is not divisible by 57 . It suffices to consider
any maximal subgroups containing such centralizers. The maximal subgroup of
shape 51+6
: 2.J2 .4, which is the normalizer associated to a 5B class, is one such
+
choice.
We first show that the Sylow 5-subgroup of M is not F SZ5 .
G :=
P :=
isoP
P :=

AtlasGroup ( ” 5 ˆ ( 1 + 6 ) : 2 . J2 . 4 ” ) ; ;
SylowSubgroup (G, 5 ) ; ;
:= IsomorphismPcGroup (P ) ; ;
Image ( i s o P ) ; ;

ex := FSZtestZ (P ) ;
The proper centralizer with order divisible by 59 is still impractical to work with.
So we will use the data for P to construct the element of order 10 mentioned above.

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

17

zp := ex [ 2 ] ; ;
zp := Image ( Inver seGener a lMa pping ( i s o P ) , zp ) ; ;
C := C e n t r a l i z e r (G, zp ) ; ;
Q := SylowSubgroup (C , 2 ) ; ;
zq := F i r s t ( Center (Q) , q−>Order ( q)>1 and
S i z e ( C e n t r a l i z e r (G, zp ∗q ) ) = 1 2 0 0 0 0 0 0 ) ; ;
#This g i v e s us t h e g and c e n t r a l i z e r we want .
g := zp ∗ zq ; ;
C := C e n t r a l i z e r (G, g ) ; ;
#Reducing t h e p e r m u t a t i o n d e g r e e w i l l
#s a v e a l o t o f compu t at ion t ime l a t e r .
iso C := S m a l l e r D e g r e e P e r m u t a t i o n R e p r e s e n t a t i o n (C ) ; ;
C := Image ( iso C ) ; ;
g := Image ( isoC , g ) ; ;
zp := Image ( isoC , zp ) ; ;
zq := Image ( isoC , zq ) ; ;
#Now
P :=
isoP
P :=

proceed to con st ru ct a choice of u .
SylowSubgroup (C , 5 ) ; ;
:= IsomorphismPcGroup (P ) ; ;
Image ( i s o P ) ; ;

ex := FSIndPt (P, 5 , Image ( iso P , zp ) ) ;
up := Image ( Inver seGener a lMa pping ( i s o P ) , ex [ 1 ] ) ; ;
#D e f i n e our c h o i c e o f u .
#In t h i s case , u has o r d e r 5 0 .
u := up∗ zq ; ;
#F i n a l l y , we compute t h e c a r d i n a l i t i e s
# of the relevant s e t s .
FSZSetCards (C, u , g , 5 , 7 ) ;
This final function yields [0,15000], which proves that M is not F SZ5 , as desired.

This final function call takes approximately 53 minutes to complete, while all
preceding operations can complete in about 5 minutes combined—though the conversion of C to a lower degree may take more than this, depending. The lower
degree C has degree 18, 125, but requires (slightly) more than 2 GB of memory to
acquire. This conversion can be skipped to keep the memory demands well under
2GB, but the execution time for FSZSetCards will inflate to approximately a day
and a half.

18

MARC KEILBERG

Remark 4.4. In the first definition of C above, containing the full Sylow 5-subgroup
of M , we have |C| = 9.45 × 1010 = 28 · 33 · 59 · 7. For the second definition of C,
corresponding to the centralizer of an element of order 10, we have |C| = 1.2×107 =
28 · 3 · 56 . The first centralizer is thus 7875 = 32 · 53 · 7 times larger than the second
one. Either one is many orders of magnitude smaller than |M | ≈ 8.1 × 1053 , but
the larger one was still too large to work with for practical purposes.
4.4. The Baby Monster. We can now consider the Baby Monster B.
Theorem 4.5. The Baby Monster B and its Sylow 5-subgroup are both non-F SZ5 .
Proof. The Baby Monster is well known to have a maximal subgroup of the form
HN.2, so it follows that B and HN have isomorphic Sylow 5-subgroups. By Theorem 4.2 HN has a non-F SZ5 Sylow 5-subgroup, so this immediately gives the
claim about the Sylow 5-subgroup of B.
From the character table of B we see that there is a unique non-identity conjugacy class whose centralizer has order divisible by 56 . This corresponds to an element of order 5 from the 5B class, and the centralizer has order 6,000,000 = 27 ·3·56 .
In the double cover 2.B of B, this centralizer is covered by the centralizer of an
element of order 10. This centralizer necessarily has order 12, 000, 000. Since M
contains 2.B as a maximal subgroup, and there is a unique centralizer of an element
of order 10 in M with order divisible by 12,000,000, these centralizers in 2.B and M
are isomorphic. We have already computed this centralizer in M in Theorem 4.3.
To obtain the centralizer in B, we need only quotient by an appropriate central
involution. In the notation of the proof of Theorem 4.3, this involution is precisely
zq.
GAP will automatically convert this quotient group D into a lower degree representation, yielding a permutation representation of degree 3125 for the centralizer.
This will require as much as 8GB of memory to complete. Moreover, the image
of zp from Theorem 4.3 in this quotient group yields the representative of the 5B
class we desire, denoted here by g. Using the image of up in the quotient for u,
we can then easily run FSZSetCards(C,u,g,5,2) to get a result of [15000,3125],
which shows that B is non-F SZ5 as desired. This final call completes in about 4
minutes.

Note that in M the final return values summed to 15,000, with one of the values
0, whereas in B they sum to 18,125 and neither is zero. This reflects how there
is no clear relationship between the F SZ properties of a group and its quotients,
even when the quotient is by a (cyclic) central subgroup. In particular, it does not
immediately follow that the quotient centralizer would yield the non-F SZ property
simply because the centralizer in M did, or vice versa.
Moreover, we also observe that the cardinalities computed in Theorem 4.2 implies that for a Sylow 5-subgroup P of B we have P5 (u, g 2 ) = ∅, so the 3125 ”extra”
elements obtained in B5 (u, g 2 ) come from non-trivial conjugates of P . This underscores the expected difficulties in a potential proof (or disproof) of Conjecture 2.7.
4.5. The Lyons group. There is exactly one other sporadic group with order
divisible by 56 (or pp+1 for p > 3): the Lyons group Ly.
Theorem 4.6. The maximal subgroup of Ly of the form 51+4 : 4.S6 has a faithful
permutation representation on 3,125 points, given by the action on the cosets of

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

19

4.S6 . Moreover, this maximal subgroup, Ly, and their Sylow 5-subgroups are all
non-F SZ5 .
Proof. It is well-known that Ly contains a copy of G2 (5) as a maximal subgroup,
and that the order of Ly is not divisible by 57 . Therefore Ly and G2 (5) have
isomorphic Sylow 5-subgroups, and by Theorem 4.1 this Sylow subgroup is not
F SZ5 .
Checking the character table for Ly as before, we find there is a unique nonidentity conjugacy class whose corresponding centralizer has order divisible by 56 .
In particular, the order of this centralizer is 2,250,000 = 24 ·32 ·56 , and it comes from
an element of order 5. So any maximal subgroup containing an element of order 5
whose centralizer has this order will suffice. The maximal subgroup 51+4 : 4.S6 is
the unique such choice.
The new difficulty here is that, by default, there are only matrix group representations available though the AtlasRep package for Ly and 51+4 : 4.S6 , which
are ill-suited for our purposes. However, faithful permutation representations for
Ly are known, and they can be constructed through GAP with sufficient memory
available provided one uses a well-chosen method. A detailed description of how to
acquire the permutation representation on 8,835,156 points, as well as downloads
for the generators (including MeatAxe versions courtesy of Thomas Breuer) can be
found on the web, courtesy Pfeiffer [12].
Using this, we can then obtain a permutation representation for the maximal subgroup 51+4 : 4.S6 on 8,835,156 points using the programs available on the online
ATLAS [16]. This in turn is fairly easily converted into a permutation representation on a much smaller number of points, provided one has up to 8 GB of memory
available, via SmallerDegreePermutationRepresentation. The author obtained
a permutation representation on 3125 points, corresponding to the action on the
cosets of 4.S6 . The exact description of the generators is fairly long, so we will not
reproduce them here. The author is happy to provide them upon request. One can
also proceed in a fashion similar to some of the cases handled in [1] to find such a
permutation representation.
Once this smaller degree representation is obtained, it is then easy to apply
the same methods as before to show the desired claims about the F SZ5 properties. We can directly compute the Sylow 5-subgroup, then find u, g through
FSZtestZ and FSIndPt irrespectively, set C to be the centralizer of g, then run
FSZSetCards(C,u,g,5,2). This returns [5000,625], which gives the desired nonF SZ5 claims.

Indeed, FSZtest can be applied to (both) the centralizer and the maximal subgroup once this permutation representation is obtained. This will complete quickly,
thanks to the relatively low orders and degrees involved. We also note that the centralizer C so obtained will not have a normal Sylow 5-subgroup, and is a perfect
group. The maximal subgroup in question is neither perfect nor solvable, and does
not have a normal Sylow 5-subgroup.
5. The F SZ sporadic simple groups
We can now show that all other sporadic simple groups and their Sylow subgroups
are F SZ.

20

MARC KEILBERG

Example 5.1. Any group which is necessarily F SZ (indeed, F SZ + ) by [4, Corollary 5.3] necessarily has all of its Sylow subgroups F SZ, and so satisfies the conjecture. This implies that all of the following sporadic groups, as well as their Sylow
p-subgroups, are F SZ (indeed, F SZ + ).
• The Mathieu groups M11 , M12 , M22 , M23 , M24 .
• The Janko groups J1 , J2 , J3 , J4 .
• The Higman-Simms group HS.
• The McLaughlin group M cL.
• The Held group He.
• The Rudvalis group Ru.
• The Suzuki group Suz.
• The O’Nan group O′ N .
• The Conway group Co3 .
• The Thompson group T h.
• The Tits group 2 F4 (2)′ .
Example 5.2. Continuing the last example, it follows that the following are the
only sporadic simple groups not immediately in compliance with the conjecture
thanks to [4, Corollary 5.3].
• The Conway groups Co1 , Co2 .
• The Fischer groups F i22 , F i23 , F i′24 .
• The Monster M .
• The Baby Monster B.
• The Lyons group Ly.
• The Harada-Norton group HN .
The previous section showed that the last four groups were all non-F SZ5 and have
non-F SZ5 Sylow 5-subgroups, and so conform to the conjecture. By exponent considerations the Sylow subgroups of the Conway and Fischer groups are all F SZ + .
The function FSZtest can be used to quickly show that Co1 , Co2 , F i22 , and F i23
are F SZ, and so conform to the conjecture.
This leaves just the largest Fischer group F i′24 .
Theorem 5.3. The sporadic simple group F i′24 and its Sylow subgroups are all
F SZ.
Proof. The exponent of F i′24 can be calculated from its character table and shown
to be
24,516,732,240 = 24 · 33 · 5 · 7 · 11 · 13 · 17 · 23 · 29.
As previously remarked, this automatically implies that the Sylow subgroups are all
F SZ (indeed, F SZ + ). By [4, Corollary 5.3] it suffices to show that every centralizer
of an element with order not in {1, 2, 3, 4, 6} in F i′24 that contains an element of
order 16 is F SZ. There is a unique conjugacy class in F i′24 for an element with
order (divisible by) 16. The centralizer of such an element has order 32, and is
isomorphic to Z16 × Z2 . So it suffices to consider the elements of order 8 in this
centralizer, and show that their centralizers (in F i′24 ) are F SZ. Every such element
has a centralizer of order 1536 = 29 · 3. So by Theorem 3.1 the result follows.
The following is GAP code verifying these claims.
G := AtlasGroup ( ” Fi2 4 ’ ” ) ; ;

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

21

GT := C h a r a c t e r T a b l e ( ” Fi2 4 ’ ” ) ; ;
P o s i t i o n s ( O r d e r s C l a s s R e p r e s e n t a t i v e s (GT) mod 1 6 , 0 ) ;
exp := Lcm( O r d e r s C l a s s R e p r e s e n t a t i v e s (GT) ) ;
C o l l e c t e d ( F a c t o r s I n t ( exp ) ) ;
SetExponent (G, exp ) ; ;
P := SylowSubgroup (G, 2 ) ; ;
#There are many ways t o g e t an e l e m e n t o f o r d e r 1 6 .
#Here ’ s a v e r y crude , i f non−d e t e r m i n i s t i c , one .
x := Random(P ) ; ;
while not Order ( x ) = 16 do x:=Random(P ) ; od ;
C := C e n t r a l i z e r (G, x ) ; ;
c e n t s := F i l t e r e d (C, y−>Order ( y ) = 8 ) ; ;
c e n t s := L i s t ( c e n t s , y−>C e n t r a l i z e r (G, y ) ) ; ;
L i s t ( cents , Size ) ;

The following then summarizes our results on sporadic simple groups.
Theorem 5.4. The following are equivalent for a sporadic simple group G.
(1) G is not F SZ.
(2) G is not F SZ5 .
(3) The order of G is divisible by 56 .
(4) G has a non-F SZ Sylow subgroup.
(5) The Sylow 5-subgroup of G is not F SZ5 .
Proof. Combine the results of this section and the previous one.



6. The symplectic group S6 (5)
In [13] it was mentioned that the symplectic group S6 (5) was likely to be the
second smallest non-F SZ simple group, after G2 (5). Computer calculations there
ran into issues when checking a particular centralizer, as the character table needed
excessive amounts of memory to compute. Our methods so far also place this group
at the extreme end of what’s reasonable. In principle the procedure and functions
we’ve introduced so far can decide that this group is non-F SZ in an estimated two
weeks of uninterrupted computations, and with nominal memory usage. However,
we can achieve a substantial improvement that completes the task in about 8 hours
(on two processes; 16 hours for a single process), while maintaining nominal memory
usage.
The simple yet critical observation comes from [13, Definition 3.3]. In particular,
if a ∈ Gm (u, g), then am = g implies that for all b ∈ classCG (g) (a) we have bm = g.
So while FSZSetCards acts as naively as possible and iterates over all elements of

22

MARC KEILBERG

C = CG (g), we in fact need to only iterate over the elements of those conjugacy
classes of C whose m-th power is g (or g n ). GAP can often compute the conjugacy
classes of a finite permutation or polycyclic group quickly and efficiently. So while
it is plausible that finding these conjugacy classes can be too memory intensive
for certain centralizers, there will nevertheless be centralizers for which all other
methods are too impractical for either time or memory reasons, but for which this
reduction to conjugacy classes makes both time and memory consumption a nonissue. The otherwise problematic centralizer of S6 (5) is precisely such a case, as we
will now see.
Theorem 6.1. The projective symplectic group S6 (5) and its Sylow 5-subgroup are
both non-F SZ5 .
Proof. As usual, our first task is to show that the Sylow 5-subgroup is non-F SZ5 ,
and then use the data obtained from that to attack S6 (5).
G :=
P :=
isoP
P :=

AtlasGroup ( ” S6 ( 5 ) ” ) ; ;
SylowSubgroup (G, 5 ) ; ;
:= IsomorphismPcGroup (P ) ; ;
Image ( i s o P ) ; ;

#Show P i s non−FSZ 5 , and
#g e t t h e g we need v i a FS Zt est Z
g := FSZtestZ (P ) [ 2 ] ;
#Get t h e u we need v i a FSIndPt
u := FSIndPt (P , 5 , g ) [ 1 ] ;
One can of course store the results of FSZtestZ and FSIndPt directly to see the
complete data returned, and then extract the specific data need.
We can then show that G = S6 (5) is itself non-F SZ5 by computing G5 (u, g) and
G5 (u, g 2 ) with the following code.
G :=
isoG
G :=
g :=
u :=
uinv

C e n t r a l i z e r (G, g ) ; ;
:= S m a l l e r D e g r e e P e r m u t a t i o n R e p r e s e n t a t i o n (G ) ; ;
Image ( isoG ) ; ;
Image ( isoG , g ) ; ;
Image ( isoG , u ) ; ;
:= I n v e r s e ( u ) ; ;

#Now we compute t h e c o n j u g a c y c l a s s e s
# of the c e n t r a l i z e r .
c l := C o n j u g a c y C l a s s e s (G ) ; ;
#We t h e n need o n l y c o n s i d e r t h o s e
# c l a s s e s w i t h a s u i t a b l e 5− t h power
cand1 := F i l t e r e d ( c l , x−>R e p r e s e n t a t i v e ( x)ˆ5=g ) ; ;
cand2 := F i l t e r e d ( c l , x−>R e p r e s e n t a t i v e ( x)ˆ5=g ˆ 2 ) ; ;

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

23

#There i s i n f a c t o n l y one c o n j u g a c y
# c l a s s in both cases .
Length ( cand1 ) ;
Length ( cand2 ) ;
cand1 := cand1 [ 1 ] ; ;
cand2 := cand2 [ 1 ] ; ;
#The f o l l o w i n g computes | G 5 ( u , g ) |
Number ( cand1 , x−>(x∗ uinv )ˆ5=g ) ;
#The f o l l o w i n g computes | G 5 ( u , g ˆ 2 ) |
Number ( cand2 , x−>(x∗ uinv )ˆ5=g ˆ 2 ) ;
This code shows that
|G5 (u, g)| = 1,875,000;
|G5 (u, g 2 )| = 375,000.
Therefore S6 (5) is non-F SZ5 , as desired.



The calculation of |G5 (u, g)| takes approximately 8.1 hours, and the calculation of |G5 (u, g 2 )| takes approximately 7.45 hours. The remaining calculations are
done in significantly less combined time. We note that the calculations of these
two cardinalities can be done independently, allowing each one to be calculated
simultaneously on separate GAP processes.
We also note that the centralizer in S6 (5) under consideration in the above
is itself a perfect group; is a permutation group of degree 3125 and order 29.25
billion; and has a non-normal Sylow 5-subgroup. Moreover, it can be shown that
the g we found yields the only rational class of P at which P fails to be F SZ. One
consequence of this, combined with the character table of S6 (5), is that, unlike in
the case of the Monster group, we are unable to switch to any other centralizer
with a smaller Sylow 5-subgroup to demonstrate the non-F SZ5 property.
Similarly as with the Baby Monster group, it is interesting to note that |P5 (u, g)| =
62,500 and |P5 (u, g 2 )| = 0 for P, u, g as in the proof. These cardinalities can be
quickly computed exactly as they were for S6 (5), simply restricted to P , or using the slower FSZSetCards, with the primary difference being that now there are
multiple conjugacy classes to check and sum over.
Before continuing on to the next section, where we consider small order perfect
groups available in GAP, we wish to note a curious dead-end, of sorts.
Lemma 6.2. Given u, g ∈ G with [u, g] = 1, let C = CG (g), D = CC (u), and
m ∈ N. Then a ∈ Gm (u, g) if and only if ad ∈ Gm (u, g) for some/any d ∈ D.
Proof. This is noted by Iovanov et al. [4] when introducing the concept of an F SZ +
group. It is an elementary consequence of the fact that D = CG (u, g) centralizes
both g and u by definition.

So suppose we have calculated those conjugacy classes in C whose m-th power is
g. As in the above code, we can iterate over all elements of these conjugacy classes
in order to compute |Gm (u, g)|. However, the preceding lemma shows that we could
instead partition each such conjugacy class into orbits under the D action. The

24

MARC KEILBERG

practical upshot then being that we need only consider a single element of each
orbit in order to compute |Gm (u, g)|.
In the specific case of the preceding theorem, we can show that the single conjugacy classes cand1 and cand2 both have precisely 234 million elements, and that
D is a non-abelian group of order 75,000, and is in fact the full centralizer of u in
S6 (5). Moreover the center of C is generated by g, and so has order 5. Thus in the
best-case scenario partitioning these conjugacy classes into D orbits can result in
orbits with |D/Z(C)| = 15,000 elements each. The cardinalities we computed can
also be observed to be multiples 15,000. That would constitute a reduction of more
than four orders of magnitude on the total number of elements we would need to
check. While this is a best-case scenario, since D also has index 390,000 in C it
seems very plausible that such a partition would produce a substantial reduction in
the number of elements to be checked. So provided that calculating these orbits can
be done reasonably quickly, we would expect a significant reduction in run-time.
There is a practical problem, however. The problem being that, as far as the
author can tell, there is no efficient way for GAP to actually compute this partition.
Doing so evidently requires that GAP fully enumerate and store the conjugacy class
in question. In our particular case, a conjugacy class of 234 million elements in a
permutation group of degree 3125 simply requires far too much memory—in excess
of 1.5 terabytes. As such, while the lemma sounds promising, it seems to be lacking
in significant practical use for computer calculations. It seems likely, in the author’s
mind, that any situation in which it is useful could have been handled in reasonable
time and memory by other methods. Nevertheless, the author cannot rule out the
idea as a useful tool.
7. Perfect groups of order less than 106
We now look for examples of additional non-F SZ perfect groups. The library
of perfect groups stored by GAP has most perfect groups of order less than 106 ,
with a few exceptions noted in the documentation. So we can iterate through the
available groups, of which there are 1097 at the time this paper was written. We
can use the function IMMtests from Section 3 to show that most of them are F SZ.
#Get a l l a v a i l a b l e s i z e s
G l i s t := F i l t e r e d ( S i z e s P e r f e c t G r o u p s ( ) ,
n−>N r P e r f e c t L i b r a r y G r o u p s ( n ) > 0 ) ; ;
#Get a l l a v a i l a b l e p e r f e c t g r o u p s
G l i s t := L i s t ( G l i s t ,
n−>L i s t ( [ 1 . . N r P e r f e c t L i b r a r y G r o u p s ( n ) ] ,
k−>P er fectGr o up ( IsPermGroup , n , k ) ) ) ; ;
G l i s t := F l a t ( G l i s t ) ; ;
#Remove t h e o b v i o u s l y FSZ on es
F l i s t := F i l t e r e d ( G l i s t , G−>not IMMtests (G)=true ) ; ;
This gives a list of 63 perfect groups which are not immediately dismissed as being
F SZ.

THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS

25

Theorem 7.1. Of the 1097 perfect groups of order less than 106 available through
the GAP perfect groups library, exactly 7 of them are not F SZ, all of which are
extensions of A5 . All seven of them are non-F SZ5 . Four of them have order
375,000 = 23 · 3 · 56 , and three of them have order 937,500 = 22 · 3 · 57 . Their perfect
group ids in the library are:
[375000, 2],

[375000, 8],

[375000, 9],

[937500, 3],

[937500, 4],

[937500, 5]

[375000, 11],

Proof. Continuing the preceding discussion, we can apply FSZtest to the 63 groups
in Flist to obtain the desired result. This calculation takes approximately two days
of total calculation time on the author’s computer, but can be easily split across
multiple GAP instances. Most of the time is spent on the F SZ groups of orders
375,000 and 937,500.

On the other hand, we can also consider the Sylow subgroups of all 1097 available
perfect groups, and test them for the F SZ property.
Theorem 7.2. If G is one of the 1097 perfect groups of order less than 106 available
through the GAP perfect groups library, then the following are equivalent.
(1)
(2)
(3)
(4)

G
G
G
G

is not F SZ.
has a non-F SZ Sylow subgroup.
has a non-F SZ5 Sylow 5-subgroup.
is not F SZ5 .

Proof. Most of the GAP calculations we need to perform now are quick, and the
problem is easily broken up into pieces, should it prove difficult to compute everything at once. The most memory intensive case requires about 1.7 GB to test.
With significantly more memory available than this, the cases can simply be tested
by FSZtest en masse, which will establish the result relatively quickly—a matter of
hours. We sketch the details here and leave it to the interested reader to construct
the relevant code. Recall that it is generally worthwhile to convert p-groups into
polycyclic groups in GAP via IsomorphismPcGroup.
Let Glist be constructed in GAP as before. Running over each perfect group,
we can easily construct their Sylow subgroups. We can then use IMMtests from
Section 3 to eliminate most cases. There are 256 Sylow subgroups, each from a
distinct perfect group, for which IMMtests is inconclusive; and there are exactly 4
cases where IMMtests definitively shows the non-F SZ property, which are precisely
the Sylow 5-subgroups of each of the non-F SZ perfect groups of order 375,000.
These 4 Sylow subgroups are all non-F SZ5 . We can also apply FSZtestZ to the
Sylow 5-subgroups of the non-F SZ perfect groups of order 937,500 to conclude
that they are all non-F SZ5 . All other Sylow subgroups remaining that come from
a perfect group of order less than 937,500 can be shown to be F SZ by applying
FSZtest without difficulty. Of the three remaining Sylow subgroups, one has a
direct factor of Z5 , and the other factor is easily tested and shown to be F SZ,
whence this Sylow subgroup is F SZ. This leaves two other cases, which are the
Sylow 5-subgroups of the perfect groups with ids [937500,7] and [937500,8]. The
second of these is easily shown to be FSZ by FSZtest. The first can also be

26

MARC KEILBERG

tested by FSZtest, but this is the case that requires the most memory and time—
approximately 15 minutes and the indicated 1.7 GB. In this case as well the Sylow
subgroups are F SZ. This completes the proof.

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