REGULAR ORBITS OF FINITE PRIMITIVE SOLVABLE GROUPS, III

arXiv:1612.05959v1 [] 18 Dec 2016

YONG YANG
Abstract. Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a
finite vector space V . Then G has a uniquely determined
p normal subgroup E which is a direct product of
extraspecial p-groups for various p and we denote e = |E/Z(E)|. We prove that when e = 2, 3, 4, 8, 9, 16,
G will have regular orbits on V when the corresponding vector space is not too small.

1. Introduction
Let G be a finite group and V a finite, faithful and completely reducible G-module. It is a
classical theme to study orbit structures of G acting on V . One of the most important and
natural questions about orbit structure is to establish the existence of an orbit of a certain
size. For a long time, there has been a deep interest and need to examine the size of the
largest possible orbits in linear group actions. The orbit {v g | g ∈ G} is called regular, if
CG (v) = 1 holds or equivalently the size of the orbit v G is |G|.
In [3], Pálfy and Pyber asked if it is possible to classify all pairs A, G with (|A|, |G|) = 1
such that A ≤ Aut(G) has a regular orbit on G. While the task is pretty challenge, at least
for primitive solvable linear groups, we can say something along this line.
Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively
on a finite vector space V . Then G has a uniquely determined normal subgroup
E which
p
is a direct product of extraspecial p-groups for various p. We denote e = |E/Z(E)| (an
invariant measuring the complexity of the group). It is proved in [6, Theorem 3.1] and [7,
Theorem 3.1] that if e = 5, 6, 7 or e ≥ 10 and e 6= 16, then G will have regular orbits on V .
Also, when e = 1, 2, 3, 4, 8, 9 or 16, there are examples where G has no regular orbit on V .
As suggested by the referee of the previous paper [7], it would be interesting and useful
to give more detailed analysis for all those smaller e values. In this paper, we attempt to
push the limit of our calculations. As one may expect, when the e are small, the calculation
gets more tedious and complicated. The bound we obtain here is not perfect, but hopefully
is good enough for most applications. I would hope that in the near future, by refining the
arguments or relying on the developments of the computer program [1], one might be able to
obtain a complete classifications up to group isomorphisms and representation isomorphisms.
Due to the nature of this question, when e are in those exceptional small cases, the
example where G does not have a regular orbit only happens when the sizes of the vector
spaces are relative small. In this paper, we study the remaining small cases, and show that
for e = 2, 3, 4, 8, 9, 16, G will have a regular orbit on V when the corresponding |V | is not
too small.
2. Notation and Lemmas
Notation:
2000 Mathematics Subject Classification. 20C20.
1

(1) Let G be a finite group, let S be a subset of G and let π be a set of different primes.
For each prime p, we denote SPp (S) = {hxi |So(x) = p, x ∈ S} and
S EPp (S) =
{x | o(x) =
SPp (S), EP(S) =
EPp (S) and
S p, x ∈ S}. We denote SP(S) =
EPπ (S) = p∈π EPp (S). We denote NEP(S) = |EP(S)|, NEPp (S) = |EPp (S)| and
NEPπ (S) = |EPπ (S)|.
(2) Let n be an even integer, q a power of a prime. Let V be a standard symplectic
vector space of dimension n of Fq . We use SCRSp(n, q) or SCRSp(V ) to denote the
set of all solvable subgroups of Sp(V ) which acts completely reducibly on V . We
use SIRSp(n, q) or SIRSp(V ) to denote the set of all solvable subgroups of Sp(V )
which acts irreducibly on V . Define SCRSp(n1 , q1 ) × SCRSp(n2 , q2 ) = {H × I | H ∈
SCRSp(n1 , q1 ) and I ∈ SCRSp(n2 , q2 )}.
(3) Let V be a finite vector space and let G ⊆ GL(V ). We define PC(G, V, p, i) = {x | x ∈
EPp (G) and dim(CV (x)) = i} and NPC(G, V, p, i) = |PC(G, V, p, i)|. We will drop V
in the notation when it is clear in the context.
(4) If V is a finite vector space of dimension n over GF(q), where q is a prime power, we
denote by Γ(q n ) = Γ(V ) the semi-linear group of V , i.e.,
Γ(q n ) = {x 7→ axσ | x ∈ GF(q n ), a ∈ GF(q n )× , σ ∈ Gal(GF(q n )/GF(q))},

and we define

Γ0 (q n ) = {x 7→ ax | x ∈ GF(q n ), a ∈ GF(q n )× }.

(5) We use H ≀ S to denote the wreath product of H with S where H is a group and S is
a permutation group.
(6) We use F(G) to denote the Fitting subgroup of G.
(7) Let n be a positive integer, we use Div(n) to denote the number of different prime
divisors of n. Clearly if n = 1, Div(n) = 0.
(8) Let n be a positive integer and p be a prime, we set Dp (n) = 0 if p ∤ n and Dp (n) = p
if p | n.
Definition 2.1. Suppose that a finite solvable group G acts faithfully, irreducibly and quasiprimitively
on a finite vector space V . Let F(G) be the Fitting subgroup of G and F(G) =
Q
P
,
i
=
1, . . . , m where Pi are normal pi -subgroups of G for different primes pi . Let
i i
Zi = Ω1 (Z(Pi )). We define

if pi is odd;
 Ω1 (Pi )
[Pi , G, · · · , G] if pi = 2 and [Pi , G, · · · , G] 6= 1;
Ei =
 Z
otherwise.
i

By proper reordering we may assume that
i = 1, . . . , s, 0 ≤ s ≤ m and Ei = Zi
Q Ei 6= Zi for Q
for i = s + 1, . . . , m. We define E = p si=1 Ei , Z = si=1 Zi and we define
p E i = Ei /Zi ,
Ē = E/Z. Furthermore, we define ei = |Ei /Zi | for i = 1, . . . , s and e = |E/Z|.
Theorem 2.2. Suppose that a finite solvable group G acts faithfully, irreducibly and quasiprimitively on an n-dimensional finite vector space V over finite field F of characteristic r.
We use the notation in Definition 2.1. Then every normal abelian subgroup of G is cyclic
and G has normal subgroups Z ≤ U ≤ F ≤ A ≤ G such that,
(1) F = EU is a central product where Z = E ∩ U = Z(E) and CG (F ) ≤ F ;
(2) F/U ∼
= E/Z is a direct sum of completely reducible G/F -modules;
2

(3) Ei is an extraspecial pi -group for i = 1, . . . , s and ei = pni i for some ni ≥ 1. Furthermore (ei , ej ) = 1 when i 6= j and e = e1 . . . es divides n, also gcd(r, e) = 1;
(4) A = CG (U) and G/A . Aut(U), A/F acts faithfully on E/Z;
(5) A/CA (Ei /Zi ) . Sp(2ni , pi );
(6) U is cyclic and acts fixed point freely on W where W is an irreducible submodule of
VU ;
(7) |V | = |W |eb for some integer b;
(8) G/A is cyclic and |G : A| | dim(W ). G = A when e = n;
(9) Let g ∈ G\A, assume that o(g) = t where t is a prime and let |W | = r m . Then t | m
and we can view the action of g on U as follows, U ⊆ F∗rm and g ∈ Gal(Frm : Fr ).
Proof. This is [7, Theorem 2.2].

Lemma 2.3. Suppose that a finite solvable group G acts faithfully, irreducibly and quasiprimitively on a finite vector space V . Using the notation in Theorem 2.2, we have |G| ≤
dim(W ) · |A/F | · e2 · (|W | − 1).
Proof. By Theorem 2.2, |G| = |G/A||A/F ||F | and |F | = |E/Z||U|. Since |G/A| | dim(W ),
|E/Z| = e2 and |U| | (|W | − 1), we have |G| | dim(W ) · |A/F | · e2 · (|W | − 1).

Lemma 2.4. Suppose that a finite solvable group G acts faithfully and quasi-primitively
on a finite vector space V over the field F. Let g ∈ EPs (G) and we use the notation in
Theorem 2.2.
1
(1) If g ∈ F then |CV (g)| ≤ |W | s eb .
3
(2) If g ∈ A\F then |CV (g)| ≤ |W |⌊ 4 e⌋b .
1
(3) If g ∈ A\F , s ≥ 3 and s ∤ |E|, then |CV (g)| ≤ |W |⌊ 2 e⌋b .
2
(4) If g ∈ A\F , s = 2 and s ∤ |E|, then |CV (g)| ≤ |W |⌊ 3 e⌋b .
1
(5) If g ∈ G\A then |CV (g)| ≤ |W | s eb .
Proof. (1) is a slight improvement of [7, Lemma 2.4(1)]. We only need to consider the cases
when s ≥ 3. Assume first that hgi ≤ F , hgi 6≤ U. Then s 6= r and hgi ≤ E ⊳ G, with
E an extraspecial s-group. We consider the restriction VE . It is a direct sum of faithful
irreducible E-modules, since Z(E) ≤ U acts fixed point freely on V . As dimensions of
centralizers do not change by extending the ground field and they add up in direct sums, we
can assume that V is an irreducible, faithful E-module on an algebraically closed field K. If
χ is the (Brauer) character corresponding to V , then, as r = char(K) 6= s, dimK (CV (S)) =
[χS , 1S ] = 1s dimK (V ) since χ(x) = 0 for every x ∈ E\Z(E), and (1) is proved.
(2), (3), (4) and (5) follow from [7, Lemma 2.4].

Lemma 2.5. Let G be Q
a finite solvable group and V be a finite, faithful irreducible FGni
module with dimension
Q pi where pi are different primes. F is algebraically closed and
char(F) = s where (s, pi ) = 1, E is a direct product Q
of normal extraspecial subgroups Ei ’s
2ni +1
of G and |Ei | = pi
. Define Zi = Z(Ei ) and Z = Zi . Consider x ∈ G, x is of prime
order different than the characteristic of V and x acts trivially on Z. In [2] Isaacs defined
good element. Let C/Z = CE/Z (x), in our situation, x is a good element if [x, C] = 1. We
call an element bad if it is not good. We have the following:
3

(1) If x is a good element, then we have that the Brauer character of x on V , say χ(x),
is such that |χ(x)|2 = |CE/Z (x)|.
(2) If x is a bad element, then χ(x) = 0.
Proof. By [2, Theorem 3.5].



Lemma 2.6. Suppose that a finite solvable group G acts faithfully, irreducibly and quasiprimitively on a finite vector space V over a field F. Using the notation in Theorem 2.2,
let x ∈ EPs (A\F ) and (s, charF) = 1. Let C/Z = CE/Z (x), we call x a good element if
[x, C] = 1, we call x a bad element if it is not good.
(1) Assume x is a bad element and let β = e/s, then |CV (x)| ≤ |W |βb .
(2) Assume x is a good element and |CE/Z (x)| ≤ a, let
1
β = ⌊ (e + (s − 1)a1/2 )⌋
s
βb
then |CV (x)| ≤ |W | .
(3) Assume o(x) = 2 and x is a good element, then |CE/Z (x)| is the square of an integer.
Assume now that 2 | e, then |CE/Z (x)| is the square of an even integer.

Proof. This is [6, Lemma 2.6].



Lemma 2.7. Assume G satisfies Theorem 2.2 and we
Q adopt the notation in it. Let p be a
prime and x ∈ EPp (A\F ) and assume |CE/Z (x)| = i pi mi . Define Up = gcd(|U|, p). We
have the following,
(1) NEPp (A\F ) ≤ NEPp (A/F )|F |.
Qp (A/F )|F | .
(2) NEPp (A\F ) ≤ NEP
Q pi 6=p pi
(3) NEPp (xF ) ≤ i Mi · Up where

if p = pi 6= 2;
 pi 2ni
2ni −mi
pi
if p 6= pi ;
Mi =
 mi
2
if p = pi = 2.
(4) Assume that p = 2 and x isQa good element. Define S = {y | y ∈ EP2 (xF ) and y is a
good element}, then |S| ≤ i Mi · U2 where

 pi 2ni −mi if pi 6= 2;
2mi
if pi = 2 and ni ≥ mi ;
Mi =
 2ni −mi
2
if pi = 2 and ni < mi .

Proof. (1) and (2) follow from [6, Lemma 2.7]. (3) and (4) are slight improvements of [6,
Lemma 2.7]. We only show the proof of (3) here. By the proof of [6, Lemma 2.7(3)], we
know that NEPp (xF/U) ≤ Mi . Let α ∈ A and o(α) = p, we consider NEPp (αU). Since
U ⊆ Z(A), NEPp (αU) ≤ Up and the result follows. (4) can be proved similarly.


Lemma 2.8. Let G be a finite solvable group and E a normal extraspecial subgroup of G.
Assume that |E| = p2n+1 , p is odd and define Z = Z(E), Ē = E/Z. Suppose that α ∈ G\E,
o(α) = 2 and CZ (α) = 1, then NEP2 (αE) ≤ pn+1 .

Proof. Set H = hα, Ei. Then H is a subgroup of G. Since CE (α) 6= E, we can view H
as a semi-direct product of E with α and α ∈ Aut(E). Consider Ē = E/Z, let I¯ = {ē ∈
¯ Any elements of E inverted by α is contained
Ē | α(ē) = ē−1 } and let I be the preimage of I.
4

in I. Since CI (α) = 1, I is abelian and |I| ≤ pn+1 . Assume o(αe) = 2 for e ∈ E, then e ∈ I
and thus NEP2 (αE) ≤ pn+1 .

Lemma 2.9. Let G be a finite solvable group and U a normal cyclic subgroup of G. Let
α ∈ G\U and o(α) = 2. Assume we can view the action of α on U as follows, U ⊆ F∗q2n and
α ∈ Gal(Fq2n : Fq ). Then NEP2 (αU) ≤ q n + 1.
n

Proof. Let u ∈ U and assume o(αu) = 2, then αuαu = 1, uq · u = uq

n +1

= 1.



Lemma 2.10. Let G be a finite solvable group and U a normal cyclic subgroup of G. Let
α ∈ G\U and o(α) = s. Assume we can view the action of α on U as follows, U ⊆ F∗qsn and
sn −1
α ∈ Gal(Fqsn : Fq ). Then NEPs (αU) ≤ qqn −1
.
Proof. Let u ∈ U and assume o(αu) = s, then αuαu . . . αu = 1.
n

1.

2n

This implies that αuα−1α2 uα−2 . . . uαs−1αs u = 1 and uq · uq · · · uq

(s−1)n

q sn −1

· u = u qn −1 =


Lemma 2.11. Let G be a finite group and K ∼
= S3 a normal subgroup of G. Suppose that
α ∈ G\K, o(α) = 2 then NEP2 (αK) ≤ 3.

Proof. Set H = hα, Ki. Then H is a subgroup of G. If CK (α) = K, then H ∼
= K × hαi and
NEP2 (αK) = 3. If CK (α) 6= K, we can view H as a semi-direct product of K with α and

α ∈ Aut(K). Since Aut(S3 ) ∼
= S3 , this cannot happen.

Lemma 2.12. Let G be a finite group and K ∼
= SL(2, 3) a normal subgroup of G. Suppose
that α ∈ G\K, o(α) = 2 then NEP2 (αK) ≤ 12.
Proof. Set H = hα, Ei. Then H is a subgroup of G. If CK (α) = K, then H ∼
= K × hαi
and NEP2 (αK) = 2. If CK (α) 6= K, we can view H as a semi-direct product of K with α
and α ∈ Aut(K). Since Aut(SL(2, 3)) ∼
= GL(2, 3) and the
= GL(2, 3)/Z(GL(2, 3)), H ∼
= S4 ∼
result is clear.


Lemma 2.13. Assume G satisfies Theorem 2.2 and we adopt the notation in it. We provide
P upper bound for the NEP(G\A). Since G/A is cyclic, we know that NEP(G\A) ≤
s NEPs (G\A) where s goes though all the prime divisor of |G/A|. Let s be a prime and
x ∈ EPs (G\A). We have the following,
(1) NEPs (xA) ≤ NEPs (xA/U)(|W | − 1) ≤ (s − 1)|A/F |e2(|W | − 1).
|−1
|−1
(2) NEPs (xA) ≤ NEPs (xA/U) |W|W|1/s
≤ (s − 1)|A/F |e2 |W|W|1/s
.
−1
−1
k
1/2
(3) If s = 2, e = p where p is an odd prime and p ∤ |W | − 1, then NEP2 (xA) ≤
|−1
NEP2 (xA/F )e |W|W|1/2
.
−1
(4) If s = 2, A . SL(2, 3), then NEP2 (xA) ≤ 12.
(5) If s = 2, A . S3 , then NEP2 (xA) ≤ 3.


Proof.

Lemma 2.14. We count the number of elements of prime order in GL(2, 2), SL(2, 2),
SL(2, 3). We have the following facts.
(1) GL(2, 2) ∼
= SL(2, 2) ∼
= S3 , NEP2 (S3 ) = 3 and NEP3 (S3 ) = 2.
(2) |SL(2, 3)| = 24, NEP2 (SL(2, 3)) = 1 and NEP3 (SL(2, 3)) = 8.
5

Proof. It is easy to check.



Lemma 2.15. Suppose that H is a linear group on V1 and S is a permutation group on Ω,
with |Ω| = n. V = V1 + V2 + · · · + Vn is an H ≀ S-module where Vi are copies of V1 .
(1) Let V1 = F22 , H ∼
= S2 and G = S3 ≀ S2 , then |G| = 62 · 2,
= S3 , n = 2, S ∼
NEP2 (G) = 21, NEP3 (G) = 8, NPC(G, 2, 3) = 6, NPC(G, 2, 2) = 15, NPC(G, 2, 1) =
0, NPC(G, 3, 2) = 4.
(2) Let V1 = F22 , H ∼
= S3 , n = 3, S ∼
= S3 and G = S3 ≀ S3 , then |G| = 64 , NEP2 (G) =
135, NEP3 (G) = 98, NPC(G, 2, 5) = 9, NPC(G, 2, 4) = 45, NPC(G, 2, 3) = 81,
NPC(G, 2, 2) = 0, NPC(G, 2, 1) = 0.
(3) Let V1 = F22 , H ∼
= S4 and G = S3 ≀ S4 , then |G| = 64 · 24, NEP(G) =
= S3 , n = 4, S ∼
1883, NPC(G, 2, 7) = 12, NPC(G, 2, 6) = 90, NPC(G, 2, 5) = 324, NPC(G, 2, 4) =
513, NPC(G, 2, 3) = 0, NPC(G, 2, 2) = 0, NPC(G, 2, 1) = 0.
(4) Let V1 = F22 , H ∼
= F20 and G = S3 ≀ F20 , then |G| = 65 · 20, NEP(G) =
= S3 , n = 5, S ∼
7169, NPC(G, 2, 8) = 90, NPC(G, 2, 6) = 585, NPC(G, 2, 4) = 0, NPC(G, 2, 2) = 0.
(5) Let V1 = F23 , H ∼
= SL(2, 3), n = 2, S ∼
= S2 and G = SL(2, 3) ≀ S2 , then |G| = 242 · 2,
NEP2 (G) = 27, NEP3 (G) = 80. NPC(G, 3, 3) = 16.
This follows from [6, Lemma 2.15].
Lemma 2.16. Let n be an even integer and V be a standard symplectic vector space of
dimension n of field F. Let G ∈ SCRSp(n, F).
(1) Let (n, F) = (2, F2 ), then G . S3 , |G| ≤ 6, NEP2 (G) ≤ 3, NEP3 (G) ≤ 2, NPC(G, 2, 1) ≤
3.
(2) Let (n, F) = (4, F2 ), then |G| ≤ 62 · 2, NEP2 (G) ≤ 21, NEP3 (G) ≤ 8, NEP{2,3}′ (G) ≤
4, NEP(G) ≤ 29, NPC(G, 2, 3) ≤ 6, NPC(G, 2, 2) ≤ 15, NPC(G, 2, 1) = 0, NPC(G, 3, 2) ≤
4.
(3) Let (n, F) = (6, F2 ), then |G| ≤ 64 , NEP2 (G) ≤ 135, NEP3 (G) ≤ 242, NEP{2,3}′ (G) ≤
6, NPC(G, 2, 5) ≤ 9, NPC(G, 2, 4) ≤ 45, NPC(G, 2, 3) ≤ 108, NPC(G, 2, 2) = 0,
NPC(G, 2, 1) = 0.
(4) Let (n, F) = (8, F2 ), then |G| ≤ 64 · 24, NEP(G) ≤ 1883, NPC(G, 2, 7) ≤ 12,
NPC(G, 2, 6) ≤ 90, NPC(G, 2, 5) ≤ 324, NPC(G, 2, 4) ≤ 513, NPC(G, 2, 3) = 0,
NPC(G, 2, 2) = 0, NPC(G, 2, 1) = 0.
(5) Let (n, F) = (2, F3 ), then G . SL(2, 3) and |G| ≤ 24. NEP2 (G) ≤ 1 and NEP3 (G) ≤
8.
(6) Let (n, F) = (4, F3 ), then |G| ≤ 242 · 2 and NEP(G) ≤ 107, NEP2 (G) ≤ 95,
NEP3 (G) ≤ 95, NEP5 (G) ≤ 64 and G will have no elements with other prime order.
Assume G 6. SL(2, 3) ≀ S2 , then NPC(G, 3, 3) = 0.
This follows from [6, Lemma 2.17].
3. Main Theorem
Theorem 3.1. Suppose that a finite solvable group G acts faithfully, irreducibly and quasiprimitively on a finite vector space V . By Theorem 2.2, G will have a uniquely determined
normal
subgroup E which is a direct product of extraspecial p-groups for various p and e =
p
|E/Z(E)|.
(1) Let e = 16. If |W | ≥ 7, or if |W | ≥ 3 and b ≥ 2, then G will have at least one regular
orbit on V .
6

(2) Let e = 9. If |W | ≥ 29 and |W | is a prime or when |W | ≥ 112 and |W | is a prime
power, then G will have at least one regular orbit on V . If |W | ≥ 7 and b ≥ 2, then
G will have at least one regular orbit on V .
(3) Let e = 8. If |W | ≥ 19 and |W | is a prime or when |W | ≥ 72 and |W | is a prime
power, then G will have at least one regular orbit on V . If |W | ≥ 5 and b ≥ 2, then
G will have at least one regular orbit on V .
(4) Let e = 4. If |W | ≥ 71 and |W | is a prime or when |W | ≥ 112 and |W | is a prime
power, then G will have at least one regular orbit on V . If b ≥ 2, and |W | ≥ 11 is a
single prime or |W | ≥ 52 is a prime power, then G will have at least one regular orbit
on V .
(5) Let e = 3. If |W | ≥ 43 and |W | is a prime or when |W | = 112 or when |W | ≥ 162 and
|W | is a prime power, then G will have at least one regular orbit on V . If |W | ≥ 13
and b ≥ 2, then G will have at least one regular orbit on V .
(6) Let e = 2. If |W | ≥ 23 and |W | is a prime or when |W | ≥ 172 and |W | is a prime
power, then G will have at least one regular orbit on V . If |W | ≥ 9 and b ≥ 2, then
G will have at least one regular orbit on V .
Proof. In order to show that G has at least one regular orbit on V it suffices to check that
[

P ∈SP(G)

CV (P ) < |V |.

In the following cases we will divide the set SP(G) into a union of sets Ai . Clearly
[

P ∈SP(G)

CV (P ) ≤

X [
i

CV (P ) .

P ∈Ai

We will find βi < e such that |CV (P )| ≤ |W |βib for all P ∈ Ai . We will find ai such that
|Ai | ≤ ai . Also we will find B such that |G| ≤ B. Since |V | = |W |eb it suffices to check that
X
ai · (|W |βib − 1) < |W |eb − 1.
i

We call this inequality ⋆.
Let e = 16. Thus 2 | |W | − 1 and A/F ∈ SCRSp(8, 2). |A/F | ≤ 64 · 24 by Lemma 2.16(4).
Define A1 = {hxi | x ∈ EP2 (F \U)}. Thus for all P ∈ A1 , |CV (P )| ≤ |W |8b by
Lemma 2.4(1) and we set β1 = 8. Since F = E  U and U ⊆ Z(F ), |A1 | ≤ 28 = a1 .
Define A2 = {hxi | x ∈ EP2 (A\F ), x is a good element and CĒ (x) = 26 }. Let hxi ∈ A2 .
1
Since x is a good element, then |CV (x)| ≤ |W |⌊ 2 (16+8)⌋b = |W |12b by Lemma 2.6(2). Thus for
all P ∈ A2 , |CV (P )| ≤ |W |12b and we set β2 = 12. |A2 | ≤ 90 · 22 · 2 = a2 by Lemma 2.16(4)
and Lemma 2.13(4).
Define A3 = {hxi | x ∈ EP2 (A\F ), x is a good element and CĒ (x) = 24 }. Let hxi ∈ A3 .
1
Since x is a good element, |CV (x)| ≤ |W |⌊ 2 (16+4)⌋b = |W |10b by Lemma 2.6(2). Thus for all
P ∈ A3 , |CV (P )| ≤ |W |10b and we set β3 = 10. |A3 | ≤ 513 · 24 · 2 = a3 by Lemma 2.16(4)
and Lemma 2.13(4).
Define A4 = {hxi | x ∈ EP2 (A\F ) and x is bad; or x ∈ EPs (A\F ) for all primes s ≥ 3}.
Let hxi ∈ A4 . If x ∈ EP2 (A\F ) and x is bad, then |CV (x)| ≤ |W |8b by Lemma 2.6(1)
7

and NEP2 (xF ) ≤ 28 · 2 by Lemma 2.13(3). If x ∈ EPs (A\F ) for a prime s ≥ 3, then
|CV (x)| ≤ |W |8b by Lemma 2.4(3) and NEPs (xF ) ≤ 28 · s by Lemma 2.13(3). Thus for all
P ∈ A4 , |CV (P )| ≤ |W |8b and we set β4 = 8. |A4 | ≤ 1883 · 28 · 2 = a4 by Lemma 2.16(4).
Define A5 = {hxi | x ∈ EP(G\A)}. Thus for all P ∈ A5 , |CV (P )| ≤ |W |8b by Lemma 2.4(5)
and we set β5 = 8. We set a5 = Div(dim(W )) · 64 · 24 · 28 · (|W |1/2 + 1) by Lemma 2.10.
It is routine to check that ⋆ is satisfied when |W | ≥ 7. ⋆ is also satisfied if |W | ≥ 3 and
b ≥ 2.
Let e = 9. Thus 3 | |W | − 1 and A/F ∈ SCRSp(4, 3). |A/F | ≤ 64 · 24 by Lemma 2.16(4).
Define A1 = {hxi | x ∈ EP3 (F \U)}. Thus for all P ∈ A1 , |CV (P )| ≤ |W |3b by
Lemma 2.4(1) and we set β1 = 3. |A1 | ≤ 35 /2 = a1 .
Define A2 = {hxi | x ∈ EP2 (A\F )}. Thus for all P ∈ A2 , |CV (P )| ≤ |W |6b by
Lemma 2.4(2) and we set β2 = 6. |A2 | ≤ 95 · 34 · 2 = a2 (this could be improved by
studying the 2 elements in A/F ) by Lemma 2.16(6) and Lemma 2.13(2).
Define A3 = {hxi | x ∈ EP3 (A\F ) and |CV (x)| ≥ |W |6b }. Let x ∈ EP3 (A\F ). If x is a bad
element, then |CV (x)| ≤ |W |3b by Lemma 2.6(1). If x is a good element and |CĒ (x)| ≤ 32 ,
1
then |CV (x)| ≤ |W |⌊ 3 (9+2·3)⌋b = |W |5b by Lemma 2.6(2). Thus for all 1 6= x ∈ P ∈ A3 , x
1
is a good element,
|CĒ (x)| = |χ(x)|2 = 33 and |W |6b ≤ |CV (x)| = |W |⌊ 3 (9+χ(x)+χ(x))⌋b ≤
√
1
|W |⌊ 3 (9+2·3· 3)⌋b = |W |6b by Lemma 2.6(2) and Lemma 2.6(3) and√ we set β3 = 6. Since
√
χ(x) = 29 ± 3 2 3 i. We may choose an
|χ(x)| = 3 · 3 and χ(x) + χ(x) = 9, we know that
√
√
element z ∈ U, o(z) = 3 such that χ(xz) = − 29 ± 3 2 3 i and χ(xz 2 ) = ∓3 3i. Now we have
|CV (xz)| ≤ 1, |CV (xz 2 )| ≤ |W |3 and thus |A3 | ≤ NPC(A/F, Ē, 3, 3)·|F/U|/2 ≤ 16·34/2 = a3
by Lemma 2.16(6).
Define A4 = {hxi | x ∈ EP3 (A\F ) and |CV (x)| ≤ |W |5}. We set β4 = 6. Also |A4 | ≤
95 · 35 /2 = a4 by Lemma 2.16(6) and Lemma 2.13(3).
Define A5 = {hxi | x ∈ EPs (H ∩ (G\A))}. Thus for all P ∈ A5 , |CV (P )| ≤ |W |4.5b by
Lemma 2.4(4) and we set β5 = 4.5. |A5 | ≤ Div(dim(W ) · 2 · 242 · 34 · (|W | − 1)) = a5 by
Lemma 2.16(6). For 2 we have |A5 | ≤ Div(dim(W )) · 2 · 242 · 34 · (|W |1/2 + 1) = a5 by
Lemma 2.16(6), Lemma 2.10 and Lemma 2.8.
It is routine to check that ⋆ is satisfied when |W | ≥ 29 and |W | is a prime or when
|W | ≥ 121 and |W | is a prime power. Also ⋆ is satisfied if |W | ≥ 7 and b ≥ 2.
Let e = 8. Thus 2 | |W | − 1 and A/F ∈ SCRSp(6, 2). |A/F | ≤ 64 by Lemma 2.16(3).
Define A1 = {hxi | x ∈ EP2 (F \U)}. Thus for all P ∈ A1 , |CV (P )| ≤ |W |4b by
Lemma 2.4(1) and we set β1 = 4. Since F = E  U and U ⊆ Z(F ), |A1 | ≤ 28 = a1 .
Define A2 = {hxi | x ∈ EP2 (A\F ), x is a good element and CĒ (x) = 24 }. Let hxi ∈ A2 .
1
Since x is a good element, |CV (x)| ≤ |W |⌊ 2 (8+4)⌋b = |W |6b by Lemma 2.6(2). Thus for
all P ∈ A2 , |CV (P )| ≤ |W |6b . |A2 | ≤ 45 · 22 · 2 by Lemma 2.16(3) and Lemma 2.13(4).
Since CĒ (x) = |χ(x)|2 = 24 , we know that χ(x) = ±4. If χ(x) = 4, then for z ∈ U,
o(z) = 2, χ(xz) = −4, and we have |CV (xz)| ≤ |W |2. Thus A2 can be split into two sets A21
and A22 . Where if hxi ∈ A21 , then |CV (x)| ≤ |W |6b and we set β21 = 6. If hxi ∈ A22 , then
|CV (x)| ≤ |W |2b and we set β22 = 2. Also |A21 | = |A22 | ≤ 45·22 and we set a21 = a22 = 45·22.

8

Define A3 = {hxi | x ∈ EP2 (A\F ) and x is bad}. Let hxi ∈ A3 , then |CV (x)| ≤ |W |4b by
Lemma 2.6(1). Thus for all P ∈ A3 , |CV (P )| ≤ |W |4b and we set β3 = 4. |A3 | ≤ 135 · 26 · 2 =
a3 by Lemma 2.16(3) and Lemma 2.13(3). need more argument
Define A4 = {hxi | x ∈ EPs (A\F ) for all primes s ≥ 3}. Let hxi ∈ A4 , then |CV (x)| ≤
|W |4b by Lemma 2.4(3). Thus for all P ∈ A4 , |CV (P )| ≤ |W |4b and we set β4 = 4.
|A4 | ≤ 248 · 26 · 3/2 = a4 by Lemma 2.16(3) and Lemma 2.13(3).
Define A5 = {hxi | x ∈ EP(G\A)}. Thus for all P ∈ A5 , |CV (P )| ≤ |W |4b by Lemma 2.4(5)
and we set β5 = 4. We set a5 = Div(dim(W )) · 64 · 26 · (|W |1/2 + 1) by Lemma 2.10.
It is routine to check that ⋆ is satisfied when |W | ≥ 19 and |W | is a prime or when
|W | ≥ 49 and |W | is a prime power. Also ⋆ is satisfied if |W | ≥ 5 and b ≥ 2.
Let e = 4. Thus 2 | |W | −1 and A/F ∈ SCRSp(4, 2) and |A/F | ≤ 62 · 2 by Lemma 2.16(2).
Define A1 = {hxi | x ∈ EP2 (F \U)}. Thus for all P ∈ A1 , |CV (P )| ≤ |W |2b by
Lemma 2.4(1) and we set β1 = 2. Since F = E  U and U ⊆ Z(F ), |A1 | ≤ 25 = a1 .
Define A2 = {hxi | x ∈ EP2 (A\F ), x is a good element and CĒ (x) = 22 }. Let hxi ∈ A2 .
1
Since x is a good element, |CV (x)| ≤ |W |⌊ 2 (4+2)⌋b = |W |3b by Lemma 2.6(2). Thus for
all P ∈ A2 , |CV (P )| ≤ |W |3b . |A2 | ≤ 15 · 22 · 2 by Lemma 2.16(2) and Lemma 2.13(4).
Since CĒ (x) = |χ(x)|2 = 22 , we know that χ(x) = ±2. If χ(x) = 2, then for z ∈ U,
o(z) = 2, χ(xz) = −2, and we have |CV (xz)| ≤ |W |. Thus A2 can be split into two sets A21
and A22 . Where if hxi ∈ A21 , then |CV (x)| ≤ |W |3b and we set β21 = 3. If hxi ∈ A22 , then
|CV (x)| ≤ |W |b and we set β22 = 3. Also |A21 | = |A22 | ≤ 15·22 and we set a21 = a22 = 15·22 .
Define A3 = {hxi | x ∈ EP2 (A\F ) and x is bad}. Let hxi ∈ A3 . Since x is a bad element,
|CV (x)| ≤ |W |2b by Lemma 2.6(1). Thus for all P ∈ A2 , |CV (P )| ≤ |W |2b and we set β3 = 2.
|A3 | ≤ 21 · 24 · 2 = a3 by Lemma 2.16(2) and Lemma 2.13(3).
Define A4 = {hxi | x ∈ EP3 (A\F )}. Let hxi ∈ A4 , then |CV (x)| ≤ |W |2b by Lemma 2.4(3).
Thus for all P ∈ A4 , |CV (P )| ≤ |W |2b and we set β4 = 2. |A4 | ≤ (4·22 +4·24)·D3 (|W |−1)/2 =
a4 by Lemma 2.16(2) and Lemma 2.13(3).
Define A5 = {hxi | x ∈ EP(G\A)}. Thus for all P ∈ A5 , |CV (P )| ≤ |W |2b by Lemma 2.4(5)
and we set β5 = 2. We set a5 = Div(dim(W )) · 2 · 62 · 24 · (|W |1/2 + 1) by Lemma 2.10.
It is routine to check that ⋆ is satisfied when |W | ≥ 71 and |W | is a prime or |W | ≥ 112
and |W | is a prime power. Also ⋆ is satisfied if b ≥ 2, |W | ≥ 11 and |W | is a prime or
|W | ≥ 52 and |W | is a prime power.
Let e = 3. Thus 2 | |W | − 1 and A/F . SL(2, 3) and |A/F | ≤ 24 by Lemma 2.16(5).
Define A1 = {hxi | x ∈ EP3 (F \U)}. Thus for all P ∈ A1 , |CV (P )| ≤ |W |b by Lemma 2.4(1)
and we set β1 = 1. Since F = E  U and U ⊆ Z(F ), |A1 | ≤ 33 /2 = a1 .
Define A2 = {hxi | x ∈ EP2 (A\F )}. Let hxi ∈ EP2 (A\F ), then |CV (x)| ≤ |W |2b by
Lemma 2.4(4). Thus for all P ∈ A2 , |CV (P )| ≤ |W |2b and we set β2 = 2. |A2 | ≤ 1 · 32 ·
D2 (|W | − 1) = a2 by Lemma 2.16(5) and Lemma 2.13(3).
Define A3 = {hxi | x ∈ EP3 (A\F )}. Let hxi ∈ EP3 (A\F ). If x is a bad element, then
|CV (x)| ≤ |W |b by Lemma 2.6(1). Let z ∈ U and o(z) = 3, by the definition in 2.6, xz and
xz 2 are all bad elements, |CV (xz)| ≤ |W |b and |CV (xz 2 )| ≤ |W |b.
If x is a good element, by the definition
in 2.6, xz and xz 2 are all good elements. Since
√
1
|CĒ (x)| = 3, |CV (x)| ≤ |W |⌊ 3 (3+2· 3)⌋ = |W |2b by Lemma 2.6(2). If |CV (x)| =√ |W |2b, then
1
1
|CĒ (x)| = |χ(x)|2 = 3 and |W |2b = |CV (x)| = |W |⌊ 3 (3+χ(x)+χ(x))⌋b ≤ |W |⌊ 3 (3+2· 3)⌋b = |W |2b
9

√
by Lemma 2.6(2)
and
Lemma
2.6(3).
Since
|χ(x)|
=
3 and χ(x) + χ(x) = 3, we know that
√
√
3
3
χ(x) = 2 ± 2 i. We may choose an element z ∈ U, o(z) = 3 such that χ(xz) = − 23 ± 23 i
√
and χ(xz 2 ) = ∓ 3i. And we have |CV (xz)| ≤ 1, |CV (xz 2 )| ≤ |W |b.
Based on the previous discussion, we can see that the worst situation is that all the
elements in A3 are good, and in this case A3 can be split into three sets A31 , A32 and
A33 . Where if hxi ∈ A31 , then |CV (x)| ≤ |W |2b and we set β31 = 2. If hxi ∈ A32 , then
|CV (x)| ≤ |W |b and we set β22 = 1. If hxi ∈ A33 , then |CV (x)| ≤ 1 and we set β33 = 0. Also
|A31 | = |A32 | = |A33 | ≤ 8 · 32 /2 and we set a31 = a32 = a33 = 8 · 32 /2 by Lemma 2.16(5) and
Lemma 2.13(3).
Define A5 = {hxi | x ∈ EP(G\A)}. Thus for all P ∈ A5 , |CV (P )| ≤ |W |1.5b by
Lemma 2.4(5) and we set β5 = 1.5. We set a4 = Div(dim(W )) · 24 · 32 · (|W | − 1). If
2, a5 = 0.5 · 24 · 32 · (|W |1/2 + 1).
It is routine to check that ⋆ is satisfied when |W | ≥ 61. |W | =
6 112 , |W | ≥ 162 and |W | is
a prime power. Also ⋆ is satisfied if |W | ≥ 13 and b ≥ 2.
Let |W | = 43, the estimate of |A3 | can be improved as the following. If A3 is empty, then
a3 = 0. Otherwise, let hxi ∈ A3 , then xE is a subgroup of A. Since 32 ∤ |W | − 1, for every
1 6= y ∈ EP3 (A\F ), y ∈ A/F · E\E. Using GAP, one checks that the number of elements of
order 3 in A/F · E\E is at most 144. 144/3 = 48 is better than 8 · 32 = 72 for a3 . The will
take care of this case. Thus the next one is |W | = 37.
Let e = 2. Thus 2 | |W | − 1, A/F . S3 and |A/F | ≤ 6 by Lemma 2.16(1).
Define A1 = {hxi | x ∈ EP2 (F \U)}. Thus for all P ∈ A1 , |CV (P )| ≤ |W |b by Lemma 2.4(1)
and we set β1 = 1. Since F = E  U and U ⊆ Z(F ), |A1 | ≤ 4 = a1 .
Define A2 = {hxi | x ∈ EP2 (A\F )}. Let hxi ∈ A2 . Since CĒ (x) = 2, x is a bad element
and |CV (x)| ≤ |W |b by Lemma 2.6(1). Thus for all P ∈ A2 , |CV (P )| ≤ |W |b and we set
β2 = 1. |A2 | ≤ 3 · 22 · 2 = a2 by Lemma 2.16(1) and Lemma 2.13(3).
Define A3 = {hxi | x ∈ EP3 (A\F )}. Let hxi ∈ A3 , then |CV (x)| ≤ |W |b by Lemma 2.4(3).
Thus for all P ∈ A3 , |CV (P )| ≤ |W |b and we set β3 = 1. |A3 | ≤ 2 · 22 · D3 (|W | − 1)/2 = a3
by Lemma 2.16(1) and Lemma 2.13(3).
Define A4 = {hxi | x ∈ EP(G\A)}. Thus for all P ∈ A4 , |CV (P )| ≤ |W |b by Lemma 2.4(5)
and we set β5 = 1. We set a4 = Div(dim(W )) · 6 · 22 · (|W |1/2 + 1) by Lemma 2.10. For if
o(x) = 2, one may reduce the estimate to 4 · 22 · (|W |1/2 + 1) by studying S3 , and reduce
further to 3 · 22 · (|W |1/2 + 1) by studying Q8  Z4 .
It is routine to check that ⋆ is satisfied when |W | ≥ 41 and |W | is a prime, or when
|W | ≥ 172 and |W | is a prime power. Also if b ≥ 2, ⋆ is satisfied when |W | ≥ 7 and |W | is a
prime, or when |W | ≥ 9 and |W | is a prime power.
And by using GAP, one may get rid of the cases when |W | = 37, 31, 29, 23 and b = 1. 

Remak: The proof of the main result is written in a way that unified decompositions and
arguments work for many different cases. In some cases, it is possible to refine the arguments
to improve the bound on |V | a little bit.
10

4. Acknowledgement
References
[1] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.3; 2002.
[2] I.M. Isaacs, ‘Characters of solvable and symplectic groups’, Amer. J. Math., Vol. 95, No. 3, Autumn (1973), 594-635.
[3] P. Pálfy and L. Pyber, ‘Small groups of automorphisms’, Bull. London Math. Soc., 30 (1998), no. 4, 386-390.
[4] O. Manz and T.R. Wolf, ‘Representations of Solvable Groups’, Cambridge University Press, 1993.
[5] A. Turull, ‘Supersolvable automorphism groups of solvable groups’, Math. Z., 183 (1983), 47-73.
[6] Y. Yang, ‘Regular orbits of finite primitive solvable groups’, J. Algebra, 323 (2010), 2735-2755.
[7] Y. Yang, ‘Regular orbits of finite primitive solvable groups, II’, J. Algebra 341 (2011), 23-34.
E-mail address: yang@txstate.edu

11

