Counting characters in linear group actions
arXiv:0704.0581v1  [math.RT]  4 Apr 2007
Counting characters in linear group actions
Thomas Michael Keller
Department of Mathematics
Texas State University
601 University Drive
San Marcos, TX 78666
e–mail: keller@txstate.edu
2000 Mathematics Subject Classification: 20C15.
http://arxiv.org/abs/0704.0581v1
Abstract. Let G be a finite group and V be a finite G–module. We present
upper bounds for the cardinalities of certain subsets of Irr(GV ), such as the set of
those χ ∈ Irr(GV ) such that, for a fixed v ∈ V , the restriction of χ to 〈v〉 is not a
multiple of the regular character of 〈v〉. These results might be useful in attacking
the non–coprime k(GV )–problem.
1 Introduction
Let G be a finite group and V be a finite G–module of characteristic p. If (|G|, |V |) = 1, then in
[3, Theorem 2.2] R. Knörr presented a beautiful argument showing how to obtain strong upper
bounds for k(GV ) (the number of conjugacy classes of GV ) by using only information on CG(v)
for a fixed v ∈ V . Note that his result immediately implies the important special case that if G
has a regular orbit on V (i.e., there is a v ∈ V with CG(v) = 1), then k(GV ) ≤ |V |, which was a
crucial result in the solution of the k(GV )–problem. In this note we give a much shorter proof
of this result (see Proposition 3.1 below).
The main objective of the paper, however, is to modify and generalize Knörr’s argument in
various directions to include non–coprime situations. This way we obtain a number of bounds
on certain subsets of Irr(GV ), such as the following:
Theorem A. Let G be a finite group and let V be a finite G–module of characteristic p. Let
v ∈ V and C = CG(v) and suppose that (|C|, |V |) = 1. Then the number of irreducible characters
whose restriction to 〈v〉 is not a multiple of the regular character of 〈v〉 is bounded above by
|CV (ci)|,
where the ci are representatives of the conjugacy classes of C.
Theorem B. Let G be a finite group and V be a finite G–module. Let g ∈ G be of prime order
not dividing |V |. Then the number of irreducible characters of GV whose restriction to A = 〈g〉
is not a multiple of the regular character of 〈g〉 is bounded above by
|CG(g)| n(CG(g), CV (g)),
where n(CG(g), CV (g)) denotes the number of orbits of CG(g) on CV (g).
Stronger versions and refinements of these results are proved in the paper. It is hoped that these
results prove useful in solving the non–coprime k(GV )–problem, as discussed, for instance, in
[2] and [1]. Theorem A and B will be proved in Sections 3 and 4 below respectively. In Section
2, we will generalize a recent result of P. Schmid [5, Theorem 2(a)] stating that in the situation
of the k(GV )–problem, if G has a regular orbit on V , then k(GV ) = |V | can only hold if G is
abelian. We prove
Theorem C. Let G be a finite group and V a finite faithful G–module with (|G|, |V |) = 1.
Suppose that G has a regular orbit on V . Then
k(GV ) ≤ |V | − |G|+ k(G).
Our proof is different from the approach taken in [5], and we actually will prove a slightly
stronger result including some non–coprime actions.
Notation: If the group A acts on the set B, we write n(A,B) for the number of orbits of A on
B. All other notation is standard or explained along the way.
2 k(GV) = |V| and regular orbits
In this paper we often work under the hypothesis of the k(GV )–problem which is the following.
2.1 Hypothesis. Let G be a finite group and let V be a finite faithful G–module such that
(|G|, |V |) = 1. Write p for the characteristic of V .
In [5, Theorem 2(a)] P. Schmid proved that under Hypothesis 2.1, if G has a regular orbit on V ,
V is irreducible, and k(GV ) = |V |, then G is abelian, and from this it follows easily that either
|G| = 1 and |V | = p, or G is cyclic of order |V | − 1. The proof in [5] is somewhat technical.
The goal of this section is to give a short proof of a generalization of Schmid’s result based on a
beautiful argument of Knörr [3]. We word it in such a way that we even do not need the coprime
hypothesis, so that the result may even be useful to study the non–coprime k(GV )–problem. To
do this, for any group X and x ∈ X we introduce the set
Irr(X,x) = {χ ∈ Irr(G)| χ|〈x〉 is not an integer multiple of the regular character of 〈x〉}
and write
k(X,x) = |Irr(X,x)|.
2.2 Theorem. Let G be a finite group and let V be a finite G–module such that G possesses a
regular orbit on V . Let v ∈ V be a representative of such an orbit. Then
k(GV, v) ≤ |V | − |G|+ k(G)
Proof. Let p be the characteristic if V . We proceed exactly as in Case (ii) of the proof of
[3, Theorem 2.2]. Write C = CG(v). As C = 1, we see that for A = 〈v〉 we trivially have that
|C| and |A| are coprime, and so that proof yields
(1) (p − 1)|V | =
τ∈Irr(GV )
(τη, τ)A
where η is the character of A defined by η = p1A − ρA with ρA being the regular character of
A. Now for any τ ∈ Irr(GV ) we have
(2) (τη, τ)A =
τ(a)(p − ρA(a))τ(a)
16=a∈A
|τ(a)|2
= 0 if τ |A is an integer multiple of ρA
≥ p− 1 otherwise
where the last step follows from [4, Corollary 4]. Next observe that if τ ∈ Irr(GV ) with V ≤ ker τ ,
then τ ∈ Irr(G) and clearly τ |A is not a multiple of ρA, and then clearly
(3) (τη, τ)A =
16=a∈A
|τ(a)|2 =
16=a∈A
τ(1)2 = (p − 1)τ(1)2.
Thus with (1), (2), and (3) we get
(p − 1)|V | =
τ∈Irr(G)
(τη, τ)A +
τ ∈ Irr(GV ),
V 6≤ ker τ
(τη, τ)A
τ∈Irr(G)
(p− 1)τ(1)2 + (k(GV, v) − k(G))(p − 1)
which yields
|V | ≥
τ∈Irr(G)
τ(1)2 + k(GV, v) − k(G) = |G|+ k(GV, v) − k(G).
This implies the assertion of the theorem, and we are done. ✸
The following consequence implies Schmid’s result [5, Theorem 2(a)].
2.3 Corollary. Assume Hypothesis 2.1 and that G has a regular orbit on V . Then
k(GV ) ≤ |V | − |G|+ k(G).
In particular, if k(GV ) = |V |, then G is abelian.
Proof. By Ito’s theorem and as (|G|, |V |) = 1, we know that χ(1) divides |G| for every
χ ∈ Irr(GV ), so in particular p does not divide χ(1). Thus for any v ∈ V # we see that
χ|〈v〉 cannot be an integer multiple of ρ〈v〉. Therefore k(GV, v) = k(GV ). Now the assertion
follows from Theorem 2.2. ✸
3 Bounds for k(GV)
In this section we study more variations of Knörr’s argument in [3, Theorem 2.2] and generalize
it to some non-coprime situations.
We begin, however, by looking at a classical application of it. An important and immediate
consequence of Knörr’s result is that if under Hypothesis 2.1 G has a regular orbit on V , then
k(GV ) ≤ |V |. This important result can be obtained in the following shorter way.
3.1 Proposition. Let G be a finite group and let V be a finite faithful G–module. Let v ∈ V .
k(GV, v) ≤ |CG(v)||V |,
in particular, if (|G|, |V |) = 1 and G has a regular orbit on V , then k(GV ) ≤ |V |.
Proof. PutA = 〈v〉. If τ ∈ Irr(GV, v), then by [4, Corollary 4] we know that
16=a∈A
|τ(a)|2 ≥ p−1.
With this and well–known character theory we get
(p− 1)k(GV, v) ≤ k(GV, v) min
τ∈Irr(GV,v)
16=a∈A
|τ(a)|2
τ∈Irr(GV )
16=a∈A
|τ(a)|2
16=a∈A
τ∈Irr(GV )
τ(a)τ(a)
16=a∈A
|CGV (a)|
16=a∈A
|CG(v)||V |
= (p− 1)|CG(v)||V |
This implies the first result. If (|G|, |V |) = 1, then by Ito’s result τ(1)||G| for all τ ∈ Irr(GV ), so
p cannot divide τ(1), and thus k(GV, v) = k(GV ), and the second result now follows by choosing
v to be in a regular orbit of G on V . ✸
Now we turn to generalizing Knörr’s argument. We discuss various ways to do so.
3.2 Remark. Let G be a finite group and let V be a finite faithful G–module of characteristic
p. Let v ∈ V and put C = CG(v) and A = 〈v〉. Let
Irr(GV,C, v) := Irr0(GV ) := Irr(GV )−{χ ∈ Irr(GV ) | χ|C×〈v〉 = τ ×ρA for a character τ of C}
Irrp′(GV ) = {χ ∈ Irr(GV ) | p does not divide χ(1)},
so that clearly Irrp′(GV ) ⊆ Irr0(GV ).
Note that if (|G|, |V |) = 1, then by Ito Irr(GV ) = Irrp′(GV ).
To work towards our next result, we again proceed somewhat similarly as in [3, Theorem 2.2]. In
the following we work under the hypothesis that (|C|, |V |) = 1. Let N = NG(A). Then |N : C|
divides p − 1. Moreover, from Knörr’s proof we know that if ci (i = 1, . . . , k(C)) with c1 = 1
are representatives of the conjugacy classes of C and aj (j = 1, . . . ,
|N :C|
) are representatives of
the N–conjugacy classes of A − 1 then, the ciaj are representatives of those conjugacy classes
of GV which intersect C × (A− 1) nontrivially.
Moreover recall from Knörr’s proof that for c ∈ C, 1 6= a ∈ A, g ∈ G, u ∈ V we know that
(ca)gu ∈ C ×A if and only if g ∈ N and u ∈ CV (c
Now define a character η on C ×A by η = 1C × (p1A − ρA).
Then for c ∈ C, a ∈ A we have
η(ca) =
p, if a 6= 1
0, if a = 1
Therefore ηGV vanishes on all conjugacy classes of GV which intersect C × (A − 1) trivially,
whereas for c ∈ C, 1 6= a ∈ A we have that
ηGV (ca) =
|C ×A|
g ∈ G
u ∈ V
η̇((ca)gu)
u∈CV (c
η(cgag)
|CV (c
|CV (c)|
= |N : C| |CV (c)|.
Thus if xi (i = 1, . . . , k(GV )) are representatives of the conjugacy classes of GV , then we get
k(GV )
ηGV (xi) =
|N:C|
ηGV (ciaj)
|N:C|
|N : C| |CV (ci)|
|N : C|
|N : C|
|CV (ci)|
= (p− 1)
|CV (ci)|,
and thus
(p − 1)
|CV (ci)| =
k(GV )
ηGV (xi) =
τ∈Irr(GV )
(τηGV , τ)GV
τ∈Irr(GV )
(τη, τ)C×A (1).
Now if τ ∈ Irr(GV ), as in [3] write
τ |C×A =
λ∈Irr(A)
τλ × λ (2)
where τλ is a character of C or τλ = 0.
Then as in [3] we see that
(τη, τ)C×A =
|C ×A|
c ∈ C
a ∈ A
τ(ca)η(ca)τ(ca)
c ∈ C
1 6= a ∈ A
τ(ca)τ(ca)
((τλ − τµ), (τλ − τµ))C (3)
where ”≤” is some arbitrary ordering on Irr(A).
Now if τλ − τµ is a nonzero multiple of ρC , then
(τλ − τµ, τλ − τµ)C ≥ |C| (4)
and thus
(τη, τ)C×A ≥ |C|.
Moreover, note that if τ ∈ Irr0(GV ), then not all τλ− τµ can be equal to 0 as otherwise from (2)
we see that τ |C×A would be equal to τλ × ρA for any λ. So we can partition the set Irr(A) into
two disjoint nonempty subsets Λ1 = {λ ∈ Irr(A) | τλ = τ1} and Λ2 = {λ ∈ Irr(A) | τλ 6= τ1},
and thus as in [3] we see that |Λ1| |Λ2| ≥ p − 1, so there are at least p − 1 pairs λ, µ ∈ Irr(A)
such that τλ − τµ 6= 0. Thus
(τη, τ)C×A ≥ p− 1 for all τ ∈ Irr0(GV ). (5)
Therefore by (1) and (5) we get that
(p− 1)
|CV (ci)| =
τ∈Irr(GV )
(τη, τ)C×A ≥
τ∈Irr0(GV )
(τη, τ)C×A ≥ (p − 1)|Irr0(GV )|
and thus
|Irr0(GV )| ≤
|CV (ci)|. (6)
From now on we assume that C > 1.
Now we repeat the arguments of this proof, but replace η by
η1 = (|C|1C − ρC)× (p1A − ρA),
so for c ∈ C and a ∈ A we have
η1(ca) =
|C|p if c 6= 1 and a 6= 1
0 if c = 1 or a = 1
Now from the above we know that the ciaj (i = 2, . . . , k(C), j = 1, . . . ,
|N :C|
) are representatives
of those conjugacy classes which intersect (C − 1)× (A− 1) nontrivially.
Clearly ηGV1 vanishes on all conjugacy classes of GV which intersect (C − 1)× (A− 1) trivially,
whereas for 1 6= c ∈ C, 1 6= a ∈ A, if (|C|, |V |) = 1, we have that
ηGV1 (ca) =
|C ×A|
g ∈ G
u ∈ V
η̇1((ca)
u∈CV (c
= |N | |CV (c)|.
Next we conclude that
k(GV )
ηGV1 (xi) =
|N:C|
ηGV1 (ciaj) = (p− 1)|C|
|CV (ci)|,
and so as in (1) we see that
(p− 1)|C|
|CV (ci)| =
τ∈Irr(GV )
(τη1, τ)C×A (7).
Now with (2) similarly as in [3] we see that
(τη1, τ)C×A =
|C ×A|
c ∈ C
a ∈ A
τ(ca)η1(ca)τ(ca)
1 6= c ∈ C
1 6= a ∈ A
τ(ca)τ(ca)
1 6= c ∈ C
1 6= a ∈ A
λ∈Irr(A)
τλ(c)λ(a)
µ∈Irr(A)
τµ(c)µ(a)
λ,µ∈Irr(A)
16=c∈C
τλ(c)τµ(c)
16=a∈A
λ(a)µ(a)
= (p − 1)
λ∈Irr(A)
16=c∈C
τλ(c)τλ(c) −
λ, µ ∈ Irr(A)
λ 6= µ
16=c∈C
τλ(c)τµ(c)
λ∈Irr(A)
16=c∈C
τλ(c)τλ(c)−
λ,µ∈Irr(A)
16=c∈C
τλ(c)τµ(c)
16=c∈C
(τλ(c) − τµ(c))(τλ(c)− τµ(c))
16=c∈C
|τλ(c) − τµ(c)|
2 (8)
for some arbitrary ordering ≤ on Irr(A).
Now recall that if τ ∈ Irr0(GV ), then not all of the τλ − τµ can be 0. So choose λ, µ ∈ Irr(C)
such that τλ − τµ 6= 0. If all the τµ (µ ∈ Irr(A)) are integer multiples of ρC then put Λ1 =
{φ ∈ Irr(A) | τφ = τλ} and Λ2 = {φ ∈ Irr(A) | τφ 6= τλ}, so Λ1 6= ∅ and Λ2 6= ∅ and from
0 ≤ (|Λ1| − 1)(|Λ2| − 1) we clearly deduce that |Λ1||Λ2| ≥ p− 1, so there are at least p− 1 pairs
(φ1, φ2) ∈ Irr(A) × Irr(A) such that τφ1 − τφ2 is a nonzero multiple of ρC .
So next we assume that τλ is not a multiple of ρC .
Then put
Γ1 = {φ ∈ Irr(A) | τλ − τφ is a multiple of ρC}
Γ2 = {φ ∈ Irr(A) | τλ − τφ is not a multiple of ρC}.
Clearly λ ∈ Γ1, so Γ1 6= ∅. If Γ2 = ∅, then Irr(A) = Γ1, and if we define Λ1, Λ2 as in the
previous argument, we see that there are at least (p − 1) pairs (φ1, φ2) ∈ Irr(A) × Irr(A) such
that τφ1 − τφ2 is a nonzero multiple of ρC .
So now suppose Γ2 6= ∅. Then |Γ1| + |Γ2| = p, and if φ1 ∈ Γ1 and φ2 ∈ Γ2, then τφ1 − τφ2 =
(τφ1 − τλ) + (τλ − τφ2) clearly is not a multiple of ρC , and by the same argument as used before
we see that |Γ1||Γ2| ≥ p − 1, so there are at least (p − 1) pairs (φ1, φ2) ∈ Irr(A) × Irr(A) such
that τφ1 − τφ2 is not a multiple of ρC .
Altogether we thus have shown that for any τ ∈ Irr0(GV ) one of the following holds:
(A) There are at least (p− 1) pairs (φ1, φ2) ∈ Irr(A)× Irr(A) such that
τφ1 − τφ2 is a nonzero multiple of ρC , or
(B) there are at least (p − 1) pairs (φ1, φ2) ∈ Irr(A)× Irr(A) such that
τφ1 − τφ2 is not a multiple of ρC .
Now it remains to consider two cases:
Case 1: At least half of the τ ∈ Irr0(GV ) satisfy (A).
Then for any of these τ by (3) and (4) we have
(τη, τ)C×A =
((τλ − τµ), (τλ − τµ))C ≥ (p − 1)|C|
and so by (1) we see that
(p − 1)
|CV (ci)| ≥
τ∈Irr0(GV )
(τη, τ)C×A ≥
|Irr0(GV )|(p− 1)|C|
which implies
|Irr0(GV )| ≤
|CV (ci)| (9).
Case 2: At least half of the τ ∈ Irr0(GV ) satisfy (B).
Then for any of these τ by (8) and [4, Corollary 4] we have
(τη1, τ)C×A ≥ (p− 1)(k(C) − 1).
Thus by (7) we have that
(p− 1)|C|
|CV (ci)| ≥
τ∈Irr0(GV )
(τη1, τ)C×A ≥
|Irr0(GV )|(p − 1) · (k(C)− 1)
whence
|Irr0(GV )| ≤
k(C)− 1
|CV (ci)| (10).
Now we drop the assumption (|C|, |V |) = 1 and work towards a general bound for |Irr0(GV )|.
For this, fix g0 ∈C such that g0 is of prime order q and put C0 = 〈g0〉 andN0 = NG(C0). Trivially
there are at most |C0|(p − 1) = q(p − 1) conjugacy classes of GV that intersect C0 × (A − 1)
nontrivially, and given 1 6= c ∈ C0, 1 6= a ∈ A, we see that for g ∈ G, u ∈ V
(ca)gu = cg[cg, u]ag ∈ C0 ×A first implies c
g ∈ C0, i.e., g ∈ N0,
and for each fixed g ∈ N0, the equation [c
g, u]ag ∈ A implies [cg, u] ∈ Aa−g which has at most
|CV (c
g)| |Ag−1| = p|CV (g0)| solutions u.
Moreover, if c = 1, then
(ca)gu = agu = ag implies g ∈ NG(A) = N and u ∈ V.
Now we define the character η2 on C0 ×A by η2 = 1C0 × (p1A − ρA). Thus η
2 vanishes on all
conjugacy classes of GV which intersect C0×(A−1) trivially, whereas for 1 6= c ∈ C0, 1 6= a ∈ A
we get
ηGV2 (ca) =
|C0 ×A|
g ∈ G
u ∈ V
η̇ ((ca)gu)
p|CV (g0)|p
|N0||CV (g0)|,
and for c = 1, 1 6= a ∈ A we get
ηGV2 (ca) = η
2 (a) =
|V |p =
|N ||V |.
Thus if xi (i = 1, . . . , k(GV )) are representatives of the conjugacy classes of GV , then
k(GV )
ηGV2 (xi) ≤ (p − 1)
|N ||V |+ (q − 1)(p − 1)
|N0||CV (g0)|
and as in (1) we see that
k(GV )
ηGV2 (xi) =
τ∈Irr(GV )
(τη2, τ)C0×A.
Now arguing as in (2), (3), (5) and (6) above will yield
|Irrp′(GV )| ≤ k(GV, v) ≤ |Irr(GV,C0, v)| ≤
(|N ||V |+ (q − 1)p|N0||CV (g0)|),
where Irr(GV,C0, v) is as defined at the beginning of Remark 3.2. Putting the main results
together, altogether we have proved the following:
3.3 Theorem. Let G be a finite group and let V be a finite faithful G–module of characteristic
p. Let v ∈ V and put C = CG(v). If ci (i = 1, . . . , k(C)) are representatives of the conjugacy
classes of C, then the following hold:
(a) If (|C|, |V |) = 1, then
|Irr0(GV )| ≤
|CV (ci)|
and if C > 1, then
|Irr0(GV )| ≤ max
|CV (ci)|,
k(C)− 1
|CV (ci)|
(b) If (|G|, |V |) = 1, then
Irr0(GV ) = Irr(G), so k(GV ) = |Irr0(GV )|
and the bounds in (a) hold true for k(GV ) instead of |Irr0(GV )|.
(c) In general, if g ∈ C such that o(g) = q is a prime, then
|Irrp′(GV )| ≤ k(GV, v) ≤
|NG(〈v〉)||V |+ (q − 1)p|NG(〈g〉)||CV (g)|
4 The dual approach
In the previous section, we always fixed v ∈ V and obtained bounds on the size of suitable subsets
of Irr(GV ) in terms of properties of the action of CG(v) on V . In this section we consider a
”dual” approach:
We fix g ∈ G and find bounds in terms of the action of CG(g) on CV (g). For this, put
Irrg(GV ) = {χ ∈ Irr(G) | χ|〈g〉×CV (g) cannot be written as ρ〈g〉×ψ for a character ψ of CV (g)}.
In particular, Irr(GV, g) ⊆ Irrg(GV ).
4.1 Theorem. Let G be a finite group and V be a finite G–module. Let g ∈ G such that
(o(g), |V |) = 1. Write A = 〈g〉, N = NG(A) and C = CV (g). Then
(a) |Irrg(GV )| ≤
(n(N,A)−1)n(CG(A),C)
(|A|−1)|C|
max16=a∈A(|NG(〈a〉)||CV (a)|)
(b) if g is of prime order, then
|Irrg(GV )| ≤ |CG(A)|n(CG(A), C)
(c) there are X,Y ⊆ Irrg(GV ) such that Irrg(GV ) is a disjoint union of X and Y and
|X| ≤
(n(N,A)− 1)n(CG(A), C)
(|A| − 1)|C|2
16=a∈A
(|NG(〈a〉)||CV (a)|) and
|Y | ≤
(n(N,A)− 1)(n(CG(A), C)− 1)
(|A| − 1)|C|
16=a∈A
(|NG(〈a〉)||CV (a)|)
(d) if g is of prime order and X,Y are as in (c), then
|X| ≤
|CG(A)|n(CG(A), C)
and |Y | ≤ |CG(A)|(n(CG(A), C)− 1)
Proof. If a1, a2 ∈ A and c1, c2 ∈ C − {1}, then it is straightforward to see that (a1, c1)
(a2, c2)
GV implies that aG1 = a
2 . Hence if T is a set of representatives of the orbits of N on
A−{1}, then every conjugacy class of GV that intersects nontrivially with (A−{1})×C has a
representative ac for some a ∈ T and some c ∈ C. Moreover, for each a ∈ T we have that if c3,
c4 ∈ C are CG(A)–conjugate, then ac3 and ac4 are CG(A)–conjugate and thus (ac3)
G = (ac4)
This shows that for each a ∈ T there are at most n(CG(A), C) conjugacy classes of GV inter-
secting nontrivially with {a} × C. Hence altogether we see that there are at most
|T |n(CG(A), C) = (n(N,A) − 1)n(CG(A), C) (1)
conjugacy classes of GV which intersect (A− {1}) × C nontrivially.
Moreover observe that for 1 6= a ∈ A, c ∈ C, h ∈ G and u ∈ V we have
(ac)hu ∈ A× C if and only if h ∈ NG(〈a〉), c
h ∈ C and u ∈ CV (a)
because the condition (ac)hu = ah[ah, u]ch ∈ A × C first forces ah ∈ A which implies (as A is
cyclic) ah ∈ 〈a〉, so h ∈ NG(〈a〉), and then as c ∈ C ≤ CV (〈a〉), it follows that c
h ∈ CV (〈a〉)
and [ah, u] ∈ [〈a〉, V ]. Now as by our hypothesis we have V = CV (〈a〉) × [〈a〉, V ], we see that
(ac)hu ∈ A× C now forces [ah, u] = 1 and ch ∈ C. Hence u ∈ CV (a
h) = CV (a).
Note that the direct product A×C is a subgroup of GV . We now define a generalized character
η on A× C by
η = (|A| · 1A − ρA)× 1C
where ρA is the regular character of A. So for a ∈ A, c ∈ C we have
η(ac) =
0, a = 1
|A|, a 6= 1
Therefore ηGV vanishes on all conjugacy classes of GV which intersect (A− {1}) × C trivially,
whereas for c ∈ C and 1 6= a ∈ A we have
ηGV (ac) =
|A× C|
h ∈ G
u ∈ V
η̇((ac)hu)
|A||C|
h ∈ NG(〈a〉)
with ch ∈ C
u∈CV (a)
η((ac)hu)
|A||C|
h ∈ NG(〈a〉)
with ch ∈ C
u∈CV (a)
η(ahch)
|CV (a)|
|A||C|
h ∈ NG(〈a〉)
with ch ∈ C
|NG(〈a〉)||CV (a)|
Thus if {xi | i = 1, . . . , k(GV )} is a set of representatives for the conjugacy classes of GV , then
by (1) and (2) we see that
(n(N,A) − 1)n(CG(A), C) ·
16=a∈A
(|NG(〈a〉)||CV (a)|) ≥
k(GV )
ηGV (xi)
τ∈Irr(GV )
(τηGV , τ)GV
τ∈Irr(GV )
(τη, τ)A×C (3).
Observe that in case that A is of prime order, then
n(N,A)− 1 =
|A| − 1
|N : CG(A)|
(|A| − 1)|CG(A)|
and max
16=a∈A
(|NG(〈a〉)||CV (a)|) = |N ||C|, so that (3) becomes
|CG(A)|(|A| − 1)n(CG(A), C) ≥
τ∈Irr(GV )
(τη, τ)A×C (3a)
Since A× C is a direct product, we can write
τA×C =
λ∈Irr(C)
(τλ × λ),
where τλ is a character of A or τλ = 0. Then
(τη, τ)A×C =
|A× C|
a ∈ A
c ∈ C
τ(ac)η(ac)τ(ac)
|A||C|
1 6= a ∈ A
c ∈ C
τ(ac)|A|τ(ac)
1 6= a ∈ A
c ∈ C
λ∈Irr(C)
τλ(a)λ(c)
µ∈Irr(C)
τµ(a)µ(c)
16=a∈A
λ,µ∈Irr(C)
τλ(a)τµ(a)
λ(c)µ(c)
16=a∈A
λ,µ∈Irr(C)
τλ(a)τµ(a)(λ, µ)C
As (λ, µ)C =
1, λ = µ
0, λ 6= µ
, we further obtain
(τη, τ)A×C =
16=a∈A
λ∈Irr(C)
τλ(a)τλ(a)
λ∈Irr(C)
16=a∈A
|τλ(a)|
2 (4)
Now observe that τ(1) =
λ∈Irr(C)
τλ(1).
If all the τλ are multiples of ρA, then clearly τ1 6∈ Irrg(GV ), and so if τ ∈ Irrg(GV ), then by [4,
Corollary 4] with (4) we see that
(τη, τ)A×C ≥ |A| − 1 (5)
So (3) and (5) yield
|Irrg(GV )| ≤
(n(N,A)− 1)n(CG(A), C)
(|A| − 1)|C|
16=a∈A
(|NG(〈a〉)||CV (a)|), (6)
and if g is of prime order, then (3a) and (5) yield
|Irrg(GV )| ≤ |CG(A)|n(CG(A), C). (6a)
Now as in Section 3, we now repeat the same arguments, but use
η1 = (|A|1A − ρA)× (|C|1C − ρC)
instead of η.
One can then easily check that
(n(N,A)− 1)(n(CG(A), C) − 1) ·
16=a∈A
(|NG(〈a〉)||CV (a)|) ≥
τ∈Irr(GV )
(τη1, τ)A×C (3b)
and if g is of prime order, then
|CG(A)|(|A| − 1)(n(CG(A), C) − 1) ≥
τ∈Irr(GV )
(τη1, τ)A×C (3c)
Moreover it is easily seen that
(τη1, τ)A×C =
1 6= a ∈ A
1 6= c ∈ C
τ(ac)τ(ac)
16=a∈A
λ,µ∈Irr(C)
τλ(a)τµ(a)
16=c∈C
λ(c)µ(c),
and as
16=c∈C
λ(c)µ(c) =
−1, if λ 6= µ
|C| − 1, if λ = µ
, it follows that
(τη1, τ)A×C =
16=a∈A
|τλ(a)− τµ(a)|
2 (7)
where ”≤” is an arbitrary ordering on Irr(C).
Next suppose that there are exactly a characters τ ∈ Irrg(GV ) such that there is a character
ψ of A (depending on τ) and there are aλ ∈ ZZ (λ ∈ Irr(C)) such that τλ = ψ + aλρA for all
λ ∈ Irr(C) and ψ is not a multiple of ρA. Then by (4) and [4, Corollary 4] we know that
(τη, τ)A×C =
λ∈Irr(C)
16=a∈A
|ψ(a)|2 ≥ |C|(|A| − 1)
and hence by (3) we get
(n(N,A)− 1)n(CG(A), C)
(|A| − 1)|C|2
16=a∈A
(|NG(〈a〉)||CV (a)|), (8)
and if g is of prime order, then by (3a) even
|CG(A)|n(CG(A), C)
Now let b be the number of τ ∈ Irrg(GV ) such that there is no such ψ.
Then there exist λ, µ ∈ Irr(C) with
16=a∈A
|τλ(a)− τµ(a)|
2 6= 0,
and thus by [4, Corollary 4] we have
(τη1, τ) ≥ |A| − 1 (9)
So (3b) and (9) yield
(n(N,A)− 1)(n(CG(A), C) − 1)
|C|(|A| − 1)
16=a∈A
(|NG(〈a〉)||CV (a)|) (10)
and, if g is of prime order, then by (3c)
b ≤ |CG(A)|(n(CG(A), C) − 1), (10b)
and clearly a+ b = |Irrg(GV )|, and hence all the assertions follow and we are done. ✸
References
[1] R. Guralnick, P. H. Tiep, The non–coprime k(GV )–problem, J. Algebra 279 (2004), 694–
[2] T. M. Keller, Fixed conjugacy classes of normal subgroups and the k(GV )–problem, J.
Algebra 305 (2006), 457–486.
[3] R. Knörr, On the number of characters in a p–block of a p–solvable group, Illinois J. Math
28 (1984), 181–209.
[4] G. R. Robinson, A bound on norms of generalized characters with applications, J. Algebra
212 (1999), 660–668.
[5] P. Schmid, Some remarks on the k(GV )–theorem, J. Group Theory 8 (2005), 589–604.
