The average character degree of finite groups and Gluck’s conjecture

Let 𝐺 be a finite group with trivial solvable radical. Then | G | is bounded from above in terms of acd ⁡ ( G ) .

Let 𝐺 be a solvable group. Then there exists λ ∈ Irr ⁡ ( F ⁢ ( G ) ) linear such that | G : F ( G ) | ≤ acd ( G | λ ) α , where α < 2.596 .

Let 𝐺 be a solvable group. Then there exists λ ∈ Irr ⁡ ( F ⁢ ( G ) ) linear such that | G : F ( G ) | ≤ acd ( G | λ ) 2 .

Let F = N 1 × ⋯ × N t , where N i is the direct product of u i ≥ 1 copies of a nonabelian simple group S i for every 𝑖. Assume that S i ≇ S j if i ≠ j . Then Aut ⁡ ( F ) ≅ Aut ⁡ ( N 1 ) × ⋯ × Aut ⁡ ( N t ) . Furthermore, Aut ⁡ ( N i ) ≅ Aut ⁡ ( S i ) ≀ S u i .

Let M , N ⁢ ⊴ ⁢ G with M ∩ N = 1 . Put K = M × N , where 𝑀 and 𝑁 are perfect. Let α ∈ Irr ⁡ ( M ) and β ∈ Irr ⁡ ( N ) be 𝐺-invariant. Assume that 𝛽 extends to 𝐺. Then α × β extends to 𝐺 if and only if 𝛼 extends to 𝐺.

Proof

Let χ ∈ Irr ⁡ ( G ) be an extension of 𝛽. Then χ K extends 𝛽 and χ ⁢ ( 1 ) = β ⁢ ( 1 ) . Since 𝑀 is perfect, χ K = 1 M × β , so β ~ = 1 M × β extends to 𝐺.

Suppose first that 𝛼 extends to 𝐺. Arguing as before, we deduce that α ~ = α × 1 N extends to 𝐺. Note that α × β = α ~ ⁢ β ~ . By [11, Corollary 6.4],

By [9, Theorem 11.7], [ β ~ ] G / K = 1 = [ α ~ ] G / K , so [ α × β ] G / K = 1 . By [9, Theorem 11.7], α × β extends to 𝐺.

Conversely, assume that α × β extends to 𝐺. Write α × β = α ~ ⁢ β ~ , where α ~ = α × 1 N is the unique extension of 𝛼 to 𝐾. By [9, Theorem 11.7] and [11, Corollary 6.4],

Using [9, Theorem 11.7] again, α ~ , and hence 𝛼, extends to 𝐺, as wanted. ∎

Let 𝑆 be a nonabelian simple group. Then | S | is bounded from above in terms of m ⁢ ( S ) .

Proof

This follows from Jordan’s theorem. Using the classification of finite simple groups, sharp bounds were obtained by M. Collins [1], building on work of Weisfeiler. ∎

Let 𝐺 be a group with trivial solvable radical. Let 𝐹 as before be the generalized Fitting subgroup of 𝐺. Let θ ∈ Irr ⁡ ( F ) and assume that θ ′ is defined. The characters 𝜃 and θ ′ satisfy the following properties.

1. θ ′ ⁢ ( 1 ) ≥ ∏ i = 1 t m ⁢ ( S i ) u i / 2 .

2. I G ⁢ ( θ ) = I G ⁢ ( θ ′ ) .

3. | Irr ⁡ ( G | θ ) | = | Irr ⁡ ( G | θ ′ ) | .

4. If γ ∈ Irr ⁡ ( F ) is such that γ ′ is defined (possibly γ = θ ), then 𝜃 and γ ′ are not 𝐺-conjugate. Furthermore, if 𝜃 and 𝛾 are not 𝐺-conjugate, then θ ′ and γ ′ are not 𝐺-conjugate.

Proof

(i) It suffices to note that θ ′ has been defined so that the number of nonprincipal irreducible factors corresponding to the copies of each S i is at least one half of the number of copies of S i .

(ii) Let Γ = Aut ⁡ ( F ) . It follows from Lemma 2.1 and the fact that the characters α i are Aut ⁡ ( S i ) -invariant that I Γ ⁢ ( θ ) = I Γ ⁢ ( θ ′ ) . Since 𝐺 is a subgroup of Γ, the result follows.

(iii) Put T = I G ⁢ ( θ ) = I G ⁢ ( θ ′ ) . By Clifford’s correspondence [9, Theorem 6.11], it suffices to see that | Irr ⁡ ( T | θ ) | = | Irr ⁡ ( T | θ ′ ) | . These numbers are known to be the number of 𝜃-special classes and the number of θ ′ -special classes, respectively [9, Problem 11.10]. By [9, Problem 11.9], it suffices to see that if H ≤ T , then 𝜃 extends to 𝐻 if and only if θ ′ extends to 𝐻.

Let H ≤ T and assume that 𝜃 extends to 𝐻. Our task is to see that θ ′ extends to 𝐻. Put N = Ker ⁡ θ . Note that 𝑁 is the direct product of the simple groups S i ⁢ j whose corresponding factor is the principal character. Let 𝑀 be the direct product of the remaining direct factors of 𝐹 so that F = M × N . Note that, since 𝜃 is 𝑇-invariant, 𝑀 and 𝑁 are normal in 𝑇. In particular, 𝑀 and 𝑁 are normal in 𝐻 too. Write θ = θ ~ × 1 N . Since 𝜃 extends to 𝐻, θ ~ extends to 𝐻, and by Lemma 2.2, so does the character obtained after the first step in the transformation from 𝜃 to θ ′ . (Note that θ ¯ = θ ~ × φ for some character 𝜑 all of whose factors are either principal characters or α i for some 𝑖. Since α i extends to Aut ⁡ ( S i ) , it follows from [18, Lemma 1.3] that 1 M × φ extends to its inertia group in Γ. In particular, 𝜑 extends to 𝐻.) If we need to replace some factors of this character θ ¯ by principal characters, then we can apply Lemma 2.2 for a second time to deduce that θ ′ extends to 𝐻.

Analogously, one can see that if θ ′ extends to 𝐻, then 𝜃 extends to 𝐻 too. This completes the proof.

(iv) Again, this follows from Lemma 2.1 and the fact that 𝐺 is a subgroup of Aut ⁡ ( F ) . ∎

Let 𝐺 be a group with trivial solvable radical. Then | G | is bounded (from above) in terms of acd ⁡ ( G ) .

Proof

Let 𝐹, as before, be the generalized Fitting subgroup of 𝐺. Since 𝐺 is isomorphic to a subgroup of Aut ⁡ ( F ) , it suffices to bound | F | . For this, we want to see that | S i | and u i is bounded in terms of acd ⁡ ( G ) for every 𝑖.

Let

be a complete system of representatives of the 𝐺-orbits on Irr ⁡ ( F ) , where θ i ′ is not defined for i ∈ { r + 1 , … , s } . Note that such a complete system of representatives exists by Lemma 2.4 (iv). We have

and these subsets form a partition of Irr ⁡ ( G ) .

Note that if χ ∈ Irr ⁡ ( G | θ j ) for some j > r , then

where we have used the inequality that precedes Lemma 2.4. On the other hand, the average degree of the irreducible characters of 𝐺 that lie over T i = { θ i , θ i ′ } (for every i ≤ r ) is

where we have used Lemma 2.4 (iii) together with Clifford theory in the first inequality and Lemma 2.4 (i) in the second inequality.

Now, we have

If we now take acd ⁡ ( G ) to be a fixed number, we deduce that 𝑡, u i and m ⁢ ( S i ) are bounded in terms of acd ⁡ ( G ) . Since | S i | is bounded in terms of m ⁢ ( S i ) by Lemma 2.3, the result follows. ∎

Suppose that a solvable group G > 1 acts faithfully and completely reducibly on a module 𝑉 of possibly mixed characteristic. Then | G | < b ⁢ ( G , V ) α .

Proof

This is [25, Theorem 3.4]. ∎

Let 𝐺 be a solvable group. Then there exists λ ∈ Irr ⁡ ( F ⁢ ( G ) ) linear such that | G : F ( G ) | ≤ acd ( G | λ ) α .

Proof

By Gaschütz’s theorem [17, Theorem 1.12], G / F ⁢ ( G ) acts faithfully and completely reducibly on V = F ⁢ ( G ) / Φ ⁢ ( G ) . By [17, Proposition 12.1], the same holds for the action of G / F ⁢ ( G ) on Irr ⁡ ( V ) . Applying Theorem 3.1 to this action, we deduce that there exists λ ∈ Irr ⁡ ( V ) such that the size of the G / F ⁢ ( G ) -orbit of 𝜆 is

By Clifford’s correspondence [9, Theorem 6.11], all the characters in Irr ⁡ ( G | λ ) are induced from irreducible characters of I G ⁢ ( λ ) . In particular, if χ ∈ Irr ⁡ ( G | λ ) , then

It follows that

as desired. ∎

Let 𝐺 be an odd order group. Then | G : F ( G ) | ≤ acd ( G | λ ) 1.643 .

Proof

Mimic the proof of Theorem 3.2 using [24, Theorem 3.2] rather than Theorem 3.1. ∎

Let G = S 3 × F p , where 𝑝 is any odd prime and F p is the Frobenius group of order ( p − 1 ) ⁢ p . We have that

Note that G = H ⁢ V . Let λ , μ ∈ Irr ⁡ ( V ) with o ⁢ ( λ ) = 3 and o ⁢ ( μ ) = p . The action of 𝐻 on Irr ⁡ ( V ) has three nontrivial orbits, with representatives λ × 1 , 1 × μ and λ × μ . Their inertia groups in 𝐻 are, respectively, C p − 1 , C 2 and 1. Since all of them are cyclic, any character in Irr ⁡ ( V ) extends to its inertia group by [9, Corollary 11.22]. Using Clifford’s correspondence and Gallagher’s theorem [9, Corollary 6.17], we deduce that 𝐺 has p − 1 irreducible characters of degree 2 lying over 𝜆, 2 irreducible characters of degree p − 1 lying over 𝜇 and one irreducible character of degree 2 ⁢ ( p − 1 ) lying over λ × μ . These are all the characters in Irr ⁡ ( G | V ) . We deduce that

but | G : F ( G ) | = 2 ( p − 1 ) is arbitrarily large. Note that Irr ⁡ ( G | V ) coincides with the set of nonlinear irreducible characters of 𝐺.

Let G = S 4 ≀ S 3 . Recall that S 4 is the semidirect product of S 3 acting faithfully and irreducibly on W = C 2 × C 2 so that V = F ⁢ ( G ) is elementary abelian of order 2 6 . Write G = H ⁢ V for some subgroup H ≅ G / F ⁢ ( G ) . As in [23, Example 13], the 𝐺-orbits in Irr ⁡ ( V ) have size 1, 9, 27 and 27. Let μ ∈ Irr ⁡ ( W ) be nonprincipal so that λ = μ × μ × μ ∈ Irr ⁡ ( V ) lies in a 𝐺-orbit of size 27. Note that I H ⁢ ( λ ) = C 2 ≀ S 3 and I G ⁢ ( λ ) = Q 8 ≀ S 3 . By [9, Problem 6.18], 𝜆 extends to its inertia subgroup in 𝐺. Using that every character of the base subgroup of I H ⁢ ( λ ) extends to its inertia subgroup, it is easy to check that I H ⁢ ( λ ) = I G ⁢ ( λ ) / V has 4 linear characters, 2 irreducible characters of degree 2 and 4 irreducible characters of degree 3. In particular, its average character degree is 2. Using Gallagher’s theorem and Clifford theory, we conclude that

Thus

as desired.

On the other hand, if φ 1 , φ 2 and φ 3 are the three nonlinear irreducible characters of S 4 (two of them of degree 3 and the other one of degree 2), then

We have that χ ⁢ ( 1 ) = 108 = b ⁢ ( G ) , while acd ⁡ ( G | λ ) = 54 . This example shows that, even in the case when Gluck’s conjecture does not follow from a large orbit theorem, there is perhaps room for improvement in the bound predicted by Gluck’s conjecture.

Let 𝐺 be a finite group and N ⁢ ⊴ ⁢ G . Is it true that acd ⁡ ( G / N ) is bounded from above in terms of acd ⁡ ( G ) ? Is it true that even acd ⁡ ( G / N ) ≤ acd ⁡ ( G ) ?

Let 𝐺 be a finite group. Is it true that | G : sol ( G ) | ≤ acd ( G ) 4 ?

Let 𝐺 be a finite group. Is it true that there exists λ ∈ Irr ⁡ ( F ⁢ ( G ) ) linear such that | G : F ( G ) | ≤ acd ( G | λ ) 4 ?

Let 𝐺 be a solvable group. Is it true that the derived length of 𝐺 is bounded in terms of acd ⁡ ( G ) ?