Normal p -complements and monomial characters
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Normal p-complements and monomial characters Xiaoyou Chen1 · Yong Yang2 Received: 2 April 2020 / Accepted: 20 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Let G be a finite group, p be a prime and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ ) = |G : ker χ |/χ (1), where ker χ is the kernel of χ . We prove in this note that if G is solvable and cod(χ ) is a p -number for every monomial character χ ∈ Irr(G), then G has a normal p-complement. Keywords Codegree of a character · Monomial character · Monolithic character · Normal p-complement Mathematics Subject Classification Primary 20C20; Secondary 20C15
1 Introduction In this note all groups under consideration are finite. We refer the reader for notation to [5] for character theory of finite groups. Let G be a group and Irr(G) be the set of irreducible characters of G. A famous theorem of Thompson in [9] states that if the degree of every nonlinear irreducible character of a group G is divisible by a prime p then G has a normal p-complement. Later, Gow and Humphreys [4] extended the theorem of Thompson: p divides the degree of every nonlinear irreducible character of G if and only if G has a normal p-complement H and H ∩ CG (P) = 1 for every Sylow p-subgroup P of G. By applying monomial characters, recently, Pang and Lu [7] showed that if G is a solvable group and p divides the degree of every nonlinear monomial irreducible character of G then G has a normal p-complement, where a
Communicated by Adrian Constantin.
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Xiaoyou Chen [email protected] Yong Yang [email protected]
1
College of Science, Henan University of Technology, Zhengzhou 450001, China
2
Department of Mathematics, Texas State University, San Marcos, TX 78666, USA
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X. Chen, Y. Yang
character χ ∈ Irr(G) is monomial if χ is induced from a linear character of some (not necessarily proper) subgroup of G. Let χ ∈ Irr(G) and write cod(χ ) =
|G : ker χ | . χ (1)
Qian, Wang and Wei defined cod(χ ) to be the codegree of the irreducible character χ of G in [8], although the name codegree of a character was first used by Chillag et al. [1] with a slightly different definition. The properties of codegrees have gained some interest in recent years. For example, the codegrees of p-groups have been studied in [2]; and the codegree analogue of Huppert’s ρ-σ conjecture has been studied in [10]. Let G be a nontrivial group and p be a prime divisor of |G|. Then by the result of Isaacs [6] we know that cod(χ ) is a p -number for every χ ∈ Irr(G) if and only if G is a p -group. We consider the codegrees of monolithic, monomial irreducible characters in this note, and we show the following, where a group G is said to be a monolith if it has a unique minimal normal subgroup and a character χ ∈ Irr(G) is monolithic if G/ ker χ is a monolith. Theorem 1.1 Let G be a p-solvable group, where p is a prime divisor of |G|. If cod(χ ) is a p -number for every monolithic, monomial χ ∈ Irr(G), then G has a normal p-comple
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