CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS

arXiv:1011.2083v4 [] 10 Mar 2016

MANOJ K. YADAV
Abstract. In 1904, Issai Schur proved the following result. If G is an arbitrary group such
that G/ Z(G) is finite, where Z(G) denotes the center of the group G, then the commutator
subgroup of G is finite. A partial converse of this result was proved by B. H. Neumann in
1951. He proved that if G is a finitely generated group with finite commutator subgroup, then
G/ Z(G) is finite. In this short note, we exhibit few arguments of Neumann, which provide
further generalizations of converse of the above mentioned result of Schur. We classify all
finite groups G such that |G/ Z(G)| = |γ2 (G)|d , where d denotes the number of elements in a
minimal generating set for G/ Z(G). Some problems and questions are posed in the sequel.

1. Introduction
In 1951, Neumann [18, Theorem 5.3] proved the following result: If the index of Z(G) in
G is finite, then γ2 (G) is finite, where Z(G) and γ2 (G) denote the center and the commutator
subgroup of G respectively. He mentioned [19, End of page 237] that this result can be obtained
from an implicit idea of Schur [23], and his proof also used Schur’s basic idea. However there is
no mention of this fact in [18] in which Schur’s paper is also cited. In this note, this result will
be termed as ‘the Schur’s theorem’. Neumann also provided a partial converse of the Schur’s
theorem [18, Corollary 5.41] as follows: If G is finitely generated by k elements and γ2 (G) is
finite, then G/ Z(G) is finite, and bounded by |G/ Z(G)| ≤ |γ2 (G)|k .
Our first motivation of writing this note is to exhibit an idea of Neumann[18, page 179] which
proves much more than what is said above on converse of the Schur’s theorem. We quote the
text here (with a minor modification in the notations):
“Let G be generated by g1 , g2 , . . . , gk . Then
Z(G) = ∩κ=k
κ=1 CG (gκ );
for, an element of G lies in the centre if and only if it (is permutable) commutes with all the
generators of G. If G is an FC-group (group whose all conjugacy classes are of finite length),
then |G : CG (gκ )| is finite for 1 ≤ κ ≤ k, and Z(G), as intersection of a finite set of subgroups
of finite index, also has finite index. The index of the intersection of two subgroups does not
exceed the product of the indices of the subgroups: hence in this case one obtains an upper bound
for the index of the centre, namely
|G : Z(G)| ≤ Πκ=k
κ=1 |G : CG (gκ )|.”

Just a soft staring at the quoted text for a moment or two suggests the following. The
conclusion does not require the group G to be FC-group. It only requires the finiteness of the
conjugacy classes of the generating elements. If a generator of the group G is contained in
Z(G), one really does not need to count it. Thus the argument works perfactly well even if G is
2010 Mathematics Subject Classification. Primary 20F24, 20E45.
Key words and phrases. commutator subgroup, Schur’s theorem, class-preserving automorphism.
1

CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS

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generated by infinite number of elements, all but finite of them lie in the center of G. Thus the
following result holds true.
Theorem A. Let G be an arbitrary group such that G/ Z(G) is finitely generated by x1 Z(G),
x2 Z(G), . . . , xt Z(G) and the conjugacy class of xi in G is of finite length for 1 ≤ i ≤ t. Then
G
G/ Z(G) is finite. Moreover |G/ Z(G)| ≤ Πti=1 |xG
i | and γ2 (G) is finite, where xi denotes the
conjugacy class of xi in G.

Neumann’s result [18, Corollary 5.41] was reproduced by Hilton [12, Theorem 1]. It seems
that Hilton was not aware of Neumann’s result. This lead two more publications [21] and [24]
dedicated to proving special cases of Theorem A.
Converse of the Schur’s theorem is not true in general as shown by infinite extraspecial pgroups, where p is an odd prime. It is interesting to know that example of such a 2-group also
exists, which is mentioned on page 238 (second para of Section 3) of [19]. It is a central product
of infinite copies of quaternion groups of order 8 amalgamated at the center of order 2.
Our second motivation of writing this note is to provide a modification of an innocent looking
result of Neumann [20, Lemma 2], which allows us to say little more on converse of the Schur’s
theorem. A modified version of this lemma is the following.
Lemma 1.1. Let G be an arbitrary group having a normal abelian subgroup A such that the
index of CG (A) in G is finite and G/A is finitely generated by g1 A, g2 A, . . . , gr A, where |giG | < ∞
for 1 ≤ i ≤ r. Then G/Z(G) is finite.
This lemma helps proving the first three statements of the following result.
Theorem B. For an arbitrary group G, G/ Z(G) is finite if any one of the following holds true:
(i) Z2 (G)/ Z(Z2 (G)) is finitely generated and γ2 (G) is finite.
(ii) G/ Z(Z2 (G)) is finitely generated and G/(Z2 (G)γ2 (G)) is finite.
(iii) γ2 (G) is finite and Z2 (G) ≤ γ2 (G).
(iv) γ2 (G) is finite and G/ Z(G) is purely non-abelian.
Our final motivation is to provide a classification of all groups G upto isoclinism (see Section
3 for the definition) such that |G/ Z(G)| = |γ2 (G)|d is finite, where d denotes the number of
elements in a minimal generating set for G/ Z(G), discuss example in various situations and pose
some problems. We conclude this section with fixing some notations. For an arbitrary group
G, by Z(G), Z2 (G) and γ2 (G) we denote the center, the second center and the commutator
subgroup of G respectively. For x ∈ G, [x, G] denotes the set {[x, g] | g ∈ G}. Notice that
|[x, G]| = |xG |, where xG denotes the conjugacy class of x in G. If [x, G] ⊆ Z(G), then [x, G]
becomes a subgroup of G. For a subgroup H of G, CG (H) denotes the centralizer of H in G
and for an element x ∈ G, CG (x) denotes the centralizer of x in G.
2. Proofs
We start with the proof of Lemma 1.1, which is essentialy same as the one given by Neumann.
Proof of Lemma 1.1. Let G/A be generated by g1 A, g2 A, . . . , gr A for some gi ∈ G, where
1 ≤ i ≤ r < ∞. Let X := {g1 , g2 , . . . , gr } and A be generated by a set Y . Then G = hX ∪ Y i
and Z(G) = CG (X) ∩ CG (Y ). Notice that CG (A) = CG (Y ). Since CG (A) is of finite index,
CG (Y ) is also of finite index in G. Also, since |giG | < ∞ for 1 ≤ i ≤ r, CG (X) is of finite index
in G. Hence the index of Z(G) in G is finite and the proof is complete.



CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS

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Proof of Theorem A can be made quite precise by using Lemma 1.1.
Proof of Theorem A. Taking A = Z(G) in Lemma 1.1, it follows that G/ Z(G) is finite.
Moreover,
|G/ Z(G)| = |G/ ∩ti=1 CG (xi )| ≤ Πti=1 |G : CG (xi )| = Πti=1 |[xi , G]| = Πti=1 |xG
i |.
That γ2 (G) is finite now follows from the Schur’s theorem.



For the proof of Theorem B we need the following result of Hall [9] and the subsequent
proposition.
Theorem 2.1. If G is an arbitrary group such that γ2 (G) is finite, then G/ Z2 (G) is finite.
Explicit bounds on the order of G/ Z2 (G) were first given by Macdonald [14, Theorem 6.2] and
later on improved by Podoski and Szegedy [22] by showing that if |γ2 (G)/(γ2 (G) ∩ Z(G))| = n,
then |G/ Z2 (G)| ≤ nc log2 n with c = 2.
Proposition 2.2. Let G be an arbitrary group such that γ2 (G) is finite and G/ Z(G) is infinite.
Then G/ Z(G) has an infinite abelian group as a direct factor.
Proof. Since γ2 (G) is finite, by Theorem 2.1 G/ Z2 (G) is finite. Thus Z2 (G)/ Z(G) is infinite.
Again using the finiteness of γ2 (G), it follows that the exponent of Z2 (G)/ Z(G) is finite. Hence
by [6, Theorem 17.2] Z2 (G)/ Z(G) is a direct sum of cyclic groups. Let G/ Z2 (G) be generated
by x1 Z2 (G), . . . , xr Z2 (G) and H := hx1 , . . . , xr i. Then it follows that modulo Z(G), H ∩ Z2 (G)
is finite. Thus we can write
Z2 (G)/ Z(G) = hy1 Z(G)i × · · · × hys Z(G)i × hys+1 Z(G)i × · · · ,
such that (H ∩ Z2 (G)) Z(G)/ Z(G) ≤ hy1 Z(G)i × · · · × hys Z(G)i. It now follows that the infinite
abelian group hys+1 Z(G)i × · · · is a direct factor of G/ Z(G), and the proof is complete.


We are now ready to prove Theorem B.
Proof of Theorem B. Since γ2 (G) is finite, it follows from Theorem 2.1 that G/ Z2 (G) is finite.
Now using the fact that Z2 (G)/ Z(Z2 (G)) is finitely generated, it follows that G/ Z(Z2 (G)) is
finitely generated. Take Z(Z2 (G)) = A. Then notice that A is a normal abelian subgroup of
G such that the index of CG (A) in G is finite, since Z2 (G) ≤ CG (A). Hence by Lemma 1.1,
G/ Z(G) is finite, which proves (i).
Again take Z(Z2 (G)) = A and notice that Z2 (G)γ2 (G) ≤ CG (A). (ii) now directly follows
from Lemma 1.1. If Z2 (G) ≤ γ2 (G), then Z2 (G) is abelian. Thus (iii) follows from (i). Finally,
(iv) follows from Proposition 2.2. This completes the proof of the theorem.



We conclude this section with an extension of Theorem A in terms of conjugacy classpreserving automorphisms of given group G. An automorphism α of an arbitrary group G
is called (conjugacy) class-preserving if α(g) ∈ g G for all g ∈ G. We denote the group of all
class-preserving automorphisms of G by Autc (G). Notice that Inn(G), the group of all inner
automorphisms of G, is a normal subgroup of Autc (G) and Autc (G) acts trivially on the center
of G. A detailed survey on class-preserving automorphisms of finite p-groups can be found in
[25].

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Let G be the group as in the statement of Theorem A. Then G is generated by x1 , x2 , . . . , xt
along with Z(G). Since Autc (G) acts trivially on the center of G, it follows that
(2.3)

| Autc (G)| ≤ Πti=1 |xG
i |

as there are only |xG
i | choices for the image of each xi under any class-preserving automorphism.
G
Since |xi | is finite for each xi , 1 ≤ i ≤ t, it follows that | Autc (G)| ≤ Πti=1 |xG
i | is finite.
We have proved the following result of which Theorem A is a corollary, because |G/ Z(G)| =
| Inn(G)| ≤ | Autc (G)|.
Theorem 2.4. Let G be an arbitrary group such that G/ Z(G) is finitely generated by x1 Z(G),
x2 Z(G), . . . , xt Z(G) and the conjugacy class of xi in G is of finite length for 1 ≤ i ≤ t. Then
Autc (G) is finite. Moreover | Autc (G)| ≤ Πti=1 |xG
i | and γ2 (G) is finite.
Proof of Theorem A is also reproduced using IA-automorphisms (automorphisms of a group
that induce identity on the abelianization) in [7, Theorem 2.3]. Proof goes on the same way as
in the case of class-preserving automorphisms.
3. Groups with maximal central quotient
We start with the following concept due to Hall [8]. For a group X, the commutator map
aX : X/ Z(X) × X/ Z(X) → γ2 (X) given by aX (x1 Z(X), x2 Z(X)) = [x1 , x2 ] is well defined.
Two groups K and H are said to be isoclinic if there exists an isomorphism φ of the factor
group K̄ = K/ Z(K) onto H̄ = H/ Z(H), and an isomorphism θ of the subgroup γ2 (K) onto
γ2 (H) such that the following diagram is commutative
a

K̄ × K̄ −−−G−→ γ2 (K)




φ×φy
yθ
a

H̄ × H̄ −−−H−→ γ2 (H).

The resulting pair (φ, θ) is called an isoclinism of K onto H. Notice that isoclinism is an
equivalence relation among groups.
The following proposition (also see Macdonald’s result [14, Lemma 2.1]) is important for the
rest of this section.
Proposition 3.1. Let G be a group such that G/ Z(G) is finite. Then there exists a finite group
H isoclinic to the group G such that Z(H) ≤ γ2 (H). Moreover if G is a p-group, then H is also
a p-group.
Proof. Let G be the given group. Then by Schur’s theorem γ2 (G) is finite. Now it follows from
a result of Hall [8] that there exists a group H which is isoclinic to G and Z(H) ≤ γ2 (H). Since
|γ2 (G)| = |γ2 (H)| is finite, Z(H) is finite. Hence, by the definition of isoclinism, H is finite.
Now suppose that G is a p-groups. Then it follows that H/ Z(H) as well as γ2 (H) are p-groups.
Since Z(H) ≤ γ2 (H), this implies that H is a p-group.



For an arbitrary group G with finite G/ Z(G), we have
(3.2)

|G/ Z(G)| ≤ |γ2 (G)|d ,

where d = d(G/ Z(G)). For simplicity we say that a group G has Property A if G/ Z(G) is finite
and equality holds in (3.2) for G. We are now going to classify, upto isoclinism, all groups G
having Property A.

CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS

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Let G be an arbitrary group having Property A. Then by Proposition 3.1 there exists a finite
group H isoclinic to G and, by the definition of isoclinism, H has Property A. Thus for classifying
all groups G, upto isoclinism, having Property A, it is sufficient to classify all finite group with
this property.
Let us first consider non-nilpotent finite groups. For such group we prove the following result
in [5]
Theorem 3.3. There is no non-nilpotent group G having Property A.
So we only need to consider finite nilpotent groups. Since a finite nilpotent group is a direct
product of it’s Sylow p-subgroups, it is sufficient to classify finite p-groups admitting Property A.
Obviously, all abelian groups admit Property A. Perhaps the simplest examples of non-abelian
groups having Property A are finite extraspecial p-groups. The class of 2-generated finite capable
nilpotent groups with cyclic commutator subgroup also admits Property A. A group G is said
∼ H/ Z(H). Isaacs [13, Theorem 2] proved:
to be capable if there exists a group H such that G =
Let G be finite and capable, and suppose that γ2 (G) is cyclic and that all elements of order 4
in γ2 (G) are central in G. Then |G/ Z(G)| ≤ |γ2 (G)|2 , and equality holds if G is nilpotent. So
G admits Property A, if G is a nilpotent group as in this statement and G is 2-generated. A
complete classification of 2-generated finite capable p-groups of class 2 is given in [15].
Motivated by finite extraspecial p-groups, a more general class of groups G admitting Property
A can be constructed as follows. For any positive integer m, let G1 , G2 , . . . , Gm be 2-generated
finite p-groups such that γ2 (Gi ) = Z(Gi ) ∼
= X (say) is cyclic of order q for 1 ≤ i ≤ m, where q
is some power of p. Consider the central product
(3.4)

Y = G1 ∗X G2 ∗X · · · ∗X Gm

of G1 , G2 , . . . , Gm amalgamated at X (isomorphic to cyclic commutator subgroups γ2 (Gi ), 1 ≤
i ≤ m). Then |Y | = q 2m+1 and |Y / Z(Y )| = q 2m = |γ2 (Y )|d(Y ) , where d(Y ) = 2m is the number
of elements in any minimal generating set for Y . Thus Y has Property A. Notice that in all of
the above examples, the commutator subgroup is cyclic. Infinite groups having Property A can
be easily obtained by taking a direct product of an infinite abelian group with any finite group
having Property A.
We now proceed to showing that any finite p-group G of class 2 having Property A is isoclinic
to a group Y defined in (3.4).
Let x ∈ Z2 (G) for a group G. Then, notice that [x, G] is a central subgroup of G. We have
the following simple but useful result.
Lemma 3.5. Let G be an arbitrary group such that Z2 (G)/ Z(G) is finitely generated by x1 Z(G),
x2 Z(G), . . . , xt Z(G) such that exp([xi , G]) is finite for 1 ≤ i ≤ t. Then
| Z2 (G)/ Z(G)| =

t
Y

exp([xi , G]).

i=1

Proof. By the given hypothesis exp([xi , G]) is finite for all i such that 1 ≤ i ≤ t. Suppose that
exp([xi , G]) = ni . Since [xi , G] ⊆ Z(G), it follows that [xni i , G] = [xi , G]ni = 1. Thus xni i ∈ Z(G)
and no smaller power of xi than ni can lie in Z(G), which implies that the order of xi Z(G) is
Qt
ni . Since Z2 (G)/ Z(G) is abelian, we have | Z2 (G)/ Z(G)| = i=1 exp([xi , G]).


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Let Φ(X) denote the Frattini subgroup of a group X. The following result provides some
structural information of p-groups of class 2 admitting Property A.
Proposition 3.6. Let H be a finite p-group of class 2 having Property A and Z(H) = γ2 (H).
Then
(i) γ2 (H) is cyclic;
(ii) H/ Z(H) is homocyclic;
(iii) [x, H] = γ2 (H) for all x ∈ H − Φ(H);
(iv) H is minimally generated by even number of elements.
Proof. Let H be the group given in the statement, which is minimally generated by d elements
x1 , x2 . . . , xd (say). Since Z(H) = γ2 (H), it follows that H/ Z(H) is minimally generated by
x1 Z(H), x2 Z(H), . . . , xd Z(H). Thus by the identity |H/ Z(H)| = |γ2 (H)|d , it follows that
order of xi Z(H) is equal to |γ2 (H)| for all 1 ≤ i ≤ d. Since the exponent of H/ Z(H) is equal
to the exponent of γ2 (H), we have that γ2 (H) is cyclic and H/ Z(H) is homocyclic. Now by
Qt
Lemma 3.5, |γ2 (H)|d = |H/ Z(H)| = i=1 exp([xi , H]). Since [xi , H] ⊆ γ2 (H), this implies that
[xi , H] = γ2 (H) for each i such that 1 ≤ i ≤ d. Let x be an arbitrary element in H − Φ(H).
Then the set {x} can always be extended to a minimal generating set of H. Thus it follows that
[x, H] = γ2 (H) for all x ∈ H − Φ(H). This proves first three assertions.
For the proof of (iv), we consider the group H̄ = H/Φ(γ2 (H)). Notice that both H as well
as H̄ are minimallay generated by d elements. Since [x, H] = γ2 (H) for all x ∈ H − Φ(H), it
follows that for no x ∈ H − Φ(H), x̄ ∈ Z(H̄), where x̄ = xΦ(γ2 (H)) ∈ H̄. Thus it follows that
Z(H̄) ≤ Φ(H̄). Also, since γ2 (H) is cyclic, γ2 (H̄) is cyclic of order p. Thus it follows that H̄ is
isoclinic to a finite extraspecial p-group, and therefore it is minimally generated by even number
of elements. Hence H is also minimally generated by even number of elements. This completes
the proof of the proposition.


Using the definition of isoclinism, we have
Corollary 3.7. Let G be a finite p-group of class 2 admitting Property A. Then γ2 (G) is cyclic
and G/ Z(G) is homocyclic.
We need the following important result.
Theorem 3.8 ([3], Theorem 2.1). Let G be a finite p-group of nilpotency class 2 with cyclic
center. Then G is a central product either of two generator subgroups with cyclic center or two
generator subgroups with cyclic center and a cyclic subgroup.
Theorem 3.9. Let G be a finite p-group of class 2 having Property A. Then G is isoclinic to
the group Y , defined in (3.4), for suitable positive integers m and n.
Proof. Let G be a group as in the statement. Then by Proposition 3.1 there exists a finite p-group
H isoclinic to G such that Z(H) = γ2 (H). Obviously H also satisfies |H/ Z(H)| = |γ2 (H)|d ,
where d denotes the number of elements in any minimal generating set of G/ Z(G). Then by
Proposition 3.6, γ2 (H) = Z(H) is cyclic of order q = pn (say) for some positive integer n, and
H/ Z(H) is homocyclic of exponent q and is of order q 2m for some positive integer m. Since
Z(H) = γ2 (H) is cyclic, it follows from Theorem 3.8 that H is a central product of 2-generated

CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS

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groups H1 , H2 , . . . , Hm . It is easy to see that γ2 (Hi ) = Z(Hi ) for 1 ≤ i ≤ m and |γ2 (H)| = q.
This completes the proof of the theorem.


We would like to remark that Theorem 3.9 is also obtained in [26, Theorem 11.2] as a
consequence on study of class-preserving automorphisms of finite p-group. But we have presented
here a direct proof.
Now we classify finite p-groups of nilpoency class larger than 2. Consider the metacylic groups
E
D
r+t
r
r+s
t
(3.10)
K := x, y | xp
= 1, y p = xp , [x, y] = xp ,
where 1 ≤ t < r and 0 ≤ s ≤ t (t ≥ 2 if p = 2) are non-negative integers. Notice that the
nilpotency class of K is at least 3. Since K is generated by 2 elements, it follows from (2.3)
that | Autc (K)| ≤ |γ2 (K)|2 = p2r . It is not so difficult to see that | Inn(K)| = |K/ Z(K)| = p2r .
Since Inn(K) ≤ Autc (K), it follows that | Autc (K)| = | Inn(K)| = |γ2 (K)|2 = |γ2 (K)|d(K) (That
Autc (G) = Inn(G), is, in fact, true for all finite metacylic p-groups). Thus K admits Property
A. Furthermore, if H is any 2-generator group isoclinic to K, then it follows that H admits
Property A. For a finite p-groups having Property A, there always exists a p-group H isoclinic
G such that |H/ Z(H)| = |γ2 (H)|d , where d = d(H). The following theorem now classifies, upto
isoclinism, all finite p-groups G of nilpotency class larger than 2 having Property A.
Theorem 3.11 (Theorem 11.3, [26]). Let G be a finite p-group of nilpotency class at least 3.
Then the following hold true.
(i) If |G/ Z(G)| = |γ2 (G)|d , where d = d(G), then d(G) = 2;
(ii) If |γ2 (G)/γ3 (G)| > 2, then |G/ Z(G)| = |γ2 (G)|d if and only if G is a 2-generator group
with cyclic commutator subgroup. Furthermore, G is isoclinic to the group K defined in (3.10)
for suitable parameters;
(iii) If |γ2 (G)/γ3 (G)| = 2, then |G/ Z(G)| = |γ2 (G)|d if and only if G is a 2-generator 2-group
of nilpotency class 3 with elementary abelian γ2 (G) of order 4.
It is clear that the groups G occuring in Theorem 3.11(iii) are isoclinic to certain groups
of order 32. Using Magma (or GAP), one can easily show that such groups of order 32 are
SmallGroup(32,k) for k = 6, 7, 8 in the small group library.
We conclude this section with providing some different type of bounds on the central quotient
of a given group. If |γ2 (G) Z(G)/ Z(G)| = n is finite for a group G, then it follows from [22,
Theorem 1] that |G/ Z2 (G)| ≤ n2 log2 n . Using this and Lemma 3.5 we can also provide an
upper bound on the size of G/ Z(G) in terms of n, the rank of Z2 (G)/ Z(G) and exponents of
certain sets of commutators (here these sets are really subgroups of G) of coset representatives
of generators of Z2 (G)/ Z(G) with the elements of G. This is given in the following theorem.
Theorem 3.12. Let G be an arbitrary group. Let |γ2 (G) Z(G)/ Z(G)| = n is finite and Z2 (G)/ Z(G)
is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) such that exp([xi , G]) is finite for 1 ≤ i ≤ t.
Then
Yt
exp([xi , G]).
|G/ Z(G)| ≤ n2 log2 n
i=1

4. Problems and Examples
Theorem B provides some conditions on a group G under which G/ Z(G) becomes finite. It
is interesting to solve

CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS

8

Problem 1. Let G be an arbitrary group. Provide a set C of optimal conditions on G such
that G/ Z(G) is finite if and only if all conditions in C hold true.
As we know that there is no finite non-nilpotent group G admitting Property A. Since
Inn(G) ≤ Autc (G), it is interesting to consider
Problem 2. Classify all non-nilpotent finite group G such that | Autc (G)| = |γ2 (G)|d , where
d = d(G).
A much stronger result than Theorem 3.3 is known in the case when the Frattini subgroup
of G is trivial. This is given in the following theorem of Herzog, Kaplan and Lev [10, Theorem
A] (the same result is also proved independently by Halasi and Podoski in [11, Theorem 1.1]).
Theorem 4.1. Let G be any non-abelian group with trivial Frattini subgroup. Then |G/ Z(G)| <
|γ2 (G)|2 .
The following result with the assertion similar to the preceding theorem is due to Isaacs [13].
Theorem 4.2. If G is a capable finite group with cyclic γ2 (G) and all elements of order 4 in
γ2 (G) are central in G, then |G : Z(G)| ≤ |γ2 (G)|2 . Moreover, equality holds if G is nilpotent.
So, there do exist nilpotent groups with comparatively small central quotient. A natural
problem is the following.
Problem 3. Classify all finite p-groups G such that |G : Z(G)| ≤ |γ2 (G)|2 .
Let G be a finite nilpotent group of class 2 minimally generated by d elements. Then it
Qd
follows from Lemma 3.5 that |G/ Z(G)| ≤ i=1 exp([xi , G]), which in turn implies
(4.3)

|G/ Z(G)| ≤ |exp(γ2 (G))|d .

Problem 4. Classify all finite p-groups G of nilpotency class 2 for which equality holds in (4.3).

Now we discuss some examples of infinite groups with finite central quotient. The most
obvious example is the infinite cyclic group. Other obvious examples are the groups G = H × Z,
where H is any finite group and Z is the infinite cyclic group. Non-obvious examples include
finitely generated F C-groups, in which conjugacy class sizes are bounded, and certain Cernikov
groups. We provide explicit examples in each case. Let Fn be the free group on n symbols and
p be a prime integer. Then the factor group Fn /(γ2 (Fn )p γ3 (Fn )) is the required group of the
first type, where γ2 (Fn )p = hup | u ∈ γ2 (Fn )i. Now let H = Z(p∞ ) × A be the direct product of
quasi-cyclic (Prüfer) group Z(p∞ ) and the cyclic group A = hai of order p, where p is a prime
integer. Now consider the group G = H ⋊ B, the semidirect product of H and the cyclic group
B = hbi of order p with the action by xb = x for all x ∈ Z(p∞ ) and ab = ac, where c is the
unique element of order p in Z(p∞ ). This group is suggested by V. Romankov and Rahul D.
Kitture through ResearchGate, and is a Cernikov group.
The following problem was suggested by R. Baer in [1].
Problem 5. Let A be an abelian group and Q be a group. Obtain necessary and sufficient
conditions on A and G so that there exists a group Q with A ∼
= Z(G) and Q ∼
= G/A.

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This problem was solved by Baer himself for an arbitrary abelian group A and finitely generated abelian group G. Moskalenko [16] solved this problem for an arbitrary abelian group A
and a periodic abelian group G. He [17] also solved this problem for arbitrary abelian group A
and a non-periodic abelian group G such that the rank of G/t(G) is 1, where t(G) denotes the
tortion subgroup of G. If this rank is more than 1, then he solved the problem when A is a
torsion abelian group. For a given group Q, the existence of a group G such that Q ∼
= G/ Z(G)
has been studied extensively under the theme ‘Capable groups’. However, to the best of our
knowledge, Problem 5 has been poorly studied in full generality. Let us restate a special case of
this problem in a little different setup. A pair of groups (A, Q), where A is an arbitray abelian
group and Q is an arbitrary group, is said to be a capable pair if there exists a group G such
that A ∼
= Z(G) and Q ∼
= G/ Z(G). So, in our situation, the following problem is very interesting.

Problem 6. Classify capable pairs (A, Q), where A is an infinite abelian group and Q is a finite
group.
Finally let us get back to the situation when G is a group with finite γ2 (G) but infinite
G/ Z(G). The well known examples of such type are infinite extraspecial p-groups. Other class
of examples can be obtained by taking a central product (amalgamated at their centers) of
infinitely many copies of a 2-generated finite p-group of class 2 such that γ2 (H) = Z(H) is cyclic
of order q, where q is some power of p. Notice that both of these classes consist of groups of
nilpotency class 2. Now if we take G = X × H, where X is an arbitrary group with finite γ2 (X)
and H is with finite γ2 (H) and infinite H/ Z(H), then γ2 (G) is finite but G/ Z(G) is infinite. So
we can construct nilpotent groups of arbitrary class and even non-nilpotent group with infinite
central quotient and finite commutator subgroup.
A non-nilpotent group G is said to be purely non-nilpotent if it does not have any non-trivial
nilpotent subgroup as a direct factor. With the help of Rahul D. Kitture, we have also been
able to construct purely non-nilpotent groups G such that γ2 (G) is finite but G/ Z(G) is infinite.
Let H be an infinite extra-special p-group. Then we can always find a field Fq , where q is
some power of a prime, containing all pth roots of unity. Now let K be the special linear group
sl(p, Fq ), which is a non-nilpotent group having a central subgroup of order p. Now consider
the group G which is a central product of H and K amalgamated at Z(H). Then G is a purely
non-nilpotent group with the required conditions. It will be interesting to see more examples of
this type which do not occur as a central product of such infinite groups of nilpotency class 2
with non-nilpotent groups.
By Proposition 2.2 we know that for an arbitrary group G with finite γ2 (G) but infinite
G/ Z(G), the group G/ Z(G) has an infinite abelian group as a direct factor. Further structural
information is highly welcome.
Problem 7. Provide structural information of the group G such that γ2 (G) is finite but G/ Z(G)
is infinite?

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DOI: 10.1007/s11856-015-1222-4
School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad
- 211 019, INDIA
E-mail address: myadav@hri.res.in

