On the sum of character degrees coprime to p and p-nilpotency of finite groups

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Abstract

The well-known Thompson theorem on character degrees states that if a prime p divides the degree of every nonlinear irreducible character of a finite group G, then G is p-nilpotent. In this paper, we give a strengthened version of Thompson's theorem in terms of χIrrp(G)χ(1) and χIrrp(G)χ(1)2, where Irrp(G) denotes the set of all ordinary irreducible characters of G of degree coprime to p.

Introduction

In the representation theory of finite groups, one of the main and important problems is to study the relationship between the character degrees of a finite group and the structure of the group. This has been a topic of interest for a long time, going back to the works in the 1960's of Isaacs-Passman [12], [13] and later in the 1980's of Huppert and his school [7].

Let G be a finite group and Irr(G) be the set of all ordinary irreducible characters of G. Let T(G):=χIrr(G)χ(1). This invariant T(G), which is often referred to as the character degree sum of G, has been known to having influence on the structure of G. In particular, in [20], [16], it was essentially shown that, the smaller |G|/T(G) is, the closer to abelian G is.

In this paper we consider a prime p and the p-version of |G|/T(G), and prove that there is a connection between this new invariant and the p-nilpotency of the group.

We writeIrrp(G):={χIrr(G)|pχ(1)} andSp(G):=χIrrp(G)χ(1)2χIrrp(G)χ(1).

Theorem A

Let G be a finite group and p be a prime. Suppose that Sp(G)<f(p), wheref(p)={2p/(p+1),ifp3,2,ifp=2. Then G is p-nilpotent.

Recall that G is said to be p-nilpotent if G has a normal subgroup H of order relatively prime to p and |G:H| is a power of p. Thompson's theorem [19, Theorem 1] on character degrees states that if p divides the degree of every nonlinear irreducible character of G, then G is p-nilpotent. Our Theorem A improves Thompson's theorem because the degree of every nonlinear irreducible character of G is divisible by p if and only if Sp(G)=1.

Of course Theorem A is not remarkable if there are no groups with 1<Sp(G)<f(p). Using GAP [4] to check all the small groups of order up to 500, we indeed found several of them. For example, there are 151 among those such that 1<S2(G)<2 and 778 among those such that 1<S3(G)<3/2.

Theorem A implies a known result that a finite group G must be nilpotent if χIrr(G)χ(1)>(2/3)|G| (see [1, Chapter 11] for instance). In fact, we can do a bit more.

Corollary B

Let G be a finite group and p the smallest odd prime divisor of |G|. Assume that χIrr(G)χ(1)>(p+1)|G|/(2p). Then G is nilpotent.

Proof

It follows from the hypothesis thatχIrr(G)χ(1)2χIrr(G)χ(1)<2pp+1<2. Note that, for every prime divisor q of |G|, every χIrr(G) of degree not coprime to q will have degree at least 2. Therefore we haveχIrr(G),q|χ(1)χ(1)2χIrr(G),q|χ(1)χ(1)2. HenceχIrr(G)χ(1)2χIrr(G),q|χ(1)χ(1)2χIrr(G)χ(1)χIrr(G),q|χ(1)χ(1)=Sq(G)χIrr(G)χ(1)2χIrr(G)χ(1). It follows that Sq(G)<2p/(p+1)f(q) for every prime divisor q of |G|. By Theorem A, we deduce that G is q-nilpotent for every q||G|, which means that G is nilpotent. 

We also find a bound for Sp(G) to ensure the solvability of G. The next result is actually needed in the proof of Theorem A.

Theorem C

Let G be a group and p be a prime. Suppose that Sp(G)<g(p), whereg(p)={15/4,ifp7,35/11,ifp=5,11/3,ifp=2,3. Then G is solvable.

The case p7 of Theorem C implies the main result of [20].

We remark that all the bounds in Theorems A and C are as tight as possible, as seen in the groups A4, D2p, A5, and SL(2,5). We also remark thatSp(G)χIrrp(G)χ(1)/|Irrp(G)| by the Cauchy-Schwarz inequality. The invariant χIrrp(G)χ(1)/|Irrp(G)|, which is often referred to as the average of p-character degrees, has been studied in [5], [14]. There, it was shown that if χIrr2(G)χ(1)/|Irr2(G)|<3/2 then G is 2-nilpotent and if χIrrp(G)χ(1)/|Irrp(G)|<2(p+1)/(p+3) for odd p then G is p-nilpotent. Consequently, if S2(G)<3/2 then G is 2-nilpotent and if Sp(G)<2(p+1)/(p+3) for odd p then G is p-nilpotent. Theorem A improves not only Thompson's theorem but this result as well.

Section snippets

Preliminaries

In this section we collect and prove some lemmas that will be needed later on.

We will write Tp(G) for χIrrp(G)χ(1). For a character χ of G, we write Irr(χ) to denote the set of irreducible constituents of χ. For NG and λIrr(N), we write IG(λ) for the inertia subgroup of λ in G, Irrp(G|λ):={χIrr(λG)|pχ(1)} and Sp(G|λ)=χIrrp(G|λ)χ(1)2/χIrrp(G|λ)χ(1). Also, we denote by nk(G) the number of irreducible characters of G of degree k. Finally we write TN to denote the semidirect

Proof of Theorem C

Theorem C will be proved in the next four propositions.

Proposition 3.1

If G is a group such that S2(G)<11/3, then G is solvable.

Proof

Suppose that G is a counterexample of minimal order. Let MG be minimal such that M is nonsolvable and let N be a minimal normal subgroup of G contained in M. Then M=M and NG. In addition, if [M,Rad(M)]1, then we choose N such that N[M,Rad(M)], where Rad(M) is the unique largest solvable normal subgroup of M, namely, the solvable radical of M.

I) Assume that N is abelian. Then G/N

Proof of Theorem A

Theorem A will be proved in the following two propositions.

Proposition 4.1

If G is a group such that S2(G)<2, then G is 2-nilpotent.

Proof

Suppose that S2(G)<2. Then we haveχIrr2(G)χ(1)(χ(1)2)<0, and so|G:G|+χIrr2(G),χ(1)3χ(1)(χ(1)2)<0. Since χ(1)(χ(1)2)2χ(1)3 for χIrr2(G) with χ(1)3, we obtain that|G:G|+χIrr2(G),χ(1)3(2χ(1)3)|G:G|+χIrr2(G),χ(1)3χ(1)(χ(1)2)<0. This implies thatχIrr2(G)(2χ(1)3)<0, and soχIrr2(G)χ(1)|Irr2(G)|<32. Now, it follows from [5, Theorem 1.1(i)] that G

Acknowledgements

The authors are grateful to the referee for his or her comments and suggestions.

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Project supported by NSF of China (Nos. 11801208, 12061011), the Jiangsu Government Scholarship for Overseas Studies (2018), the key program of NSF of Guangxi of China (No. 2020GXNSFDA238014), and a grant from the Simons Foundation (No. 499532).

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