Normal p-complements and irreducible character codegrees
aa r X i v : . [ m a t h . G R ] F e b Normal p -complements and irreduciblecharacter codegrees Jiakuan Lu, Yu Li, Boru ZhangSchool of Mathematics and Statistics, Guangxi Normal University,Guilin 541006, Guangxi, P. R. China
Abstract
Let G be a finite group and p ∈ π ( G ), and let Irr( G ) be the setof all irreducible complex characters of G . Let χ ∈ Irr( G ), we writecod( χ ) = | G : ker χ | /χ (1), and called it the codegree of the irreduciblecharacter χ . Let N E G , write Irr( G | N ) = { χ ∈ Irr( G ) | N * ker χ } ,and cod( G | N ) = { cod( χ ) | χ ∈ Irr( G | N ) } . In this Ipaper, we prove thatif N E G and every member of cod( G | N ′ ) is not divisible by some fixedprime p ∈ π ( G ), then N has a normal p -complement and N is solvable.Keywords: Finite groups; Codegree of a character; Normal p -complement.MSC(2000): 20C20, 20C15 Throughout this paper, G always denotes a finite group and, as usual, let Irr( G )be the set of all irreducible complex characters of G and cd( G ) = { χ (1) | χ ∈ Irr( G ) } . The structure of G is heavily determined by cd( G ), and there are alot of classical theorems on this subject. For example, Thompson proved thatif there is some prime p that divides every member of Irr( G ) exceeding 1, then G has a normal p -complement (See [10] or [4, Corollary 12.2]).Given that N E G , we writeIrr( G | N ) = { χ ∈ Irr( G ) | N * ker χ } and cd( G | N ) = { χ (1) | χ ∈ Irr( G | N ) } . Berkovich [1] proved a analog of the theorem of Thompson. He showed that N E G has a normal p -complement if all nonlinear members of Irr( G | N ) havedegree divisible by some fixed prime p .Isaacs and Knutson [5] proved a generalized version of Berkovich’s theorem.They proved that N E G is solvable and has a normal p -complement if everymember of cd( G | N ′ ) is divisible by p .Let χ ∈ Irr( G ). In [8], the authors definedcod( χ ) = | G : ker χ | χ (1) , and called it the codegree of the irreducible character χ . Many facts aboutcodegree of the irreducible characters in finite groups have been obtained. For1xample, see [3, 6, 8, 11]. Recently, Chen and Yang [2] proved that if G is a p -solvable group and cod( χ ) is a p ′ -number for every monolithic, monomial χ ∈ Irr( G ), then G has a normal p -complement.Let H be a maximal subgroup of G and let χ be an irreducible constituentof (1 H ) G . Following [9], we call χ a P -character of G with respect to H , anddenote by Irr P ( G ) the set of P -characters of G . In [9], Qian and Yang showedmany interesting facts about P -characters in a finite solvable group. Lu, Wuand Meng [7] proved that if G is a p -solvable group and cod( χ ) is a p ′ -numberfor every χ ∈ Irr P ( G ), then G is p -nilpotent.In this paper, we may take one more step. We writecod( G | N ) = { cod( χ ) | χ ∈ Irr( G | N ) } . Our main result is the analog of the theorem of Isaacs and Knutson.
Theorem 1.1
Let N E G and suppose that every member of cod( G | N ′ ) is notdivisible by some fixed prime p ∈ π ( G ) . Then N has a normal p -complementand N is solvable. The following result is useful in the proof of the solvability in Theorem 1.1.
Lemma 2.1 ([4, Lemma 2.2])
Suppose that A acts on G via automorphismsand that ( | A | , | G | ) = 1 . If C G ( A ) = 1 , then G is solvable. Proof of the theorem 1.1.
We first prove that N has a normal p -complement.Write M = O p ( N ), let P ∈ Syl p ( M ). Assume that P > S ∈ Syl p ( G ) such that P ≤ S . Then P = S ∩ M E S , and S permutesLin( P ), where Lin( P ) = { λ ∈ Irr( P ) | λ (1) = 1 } . Since | S | and | Lin( N ) | are p -powers, we may choose a nonprincipal linear character λ of P such that λ isstabilized by S .Now S stabilizes λ M , and thus S permutes the irreducible constituents of λ M . Since λ M (1) = | M : P | is not divisible by p , λ M must have some S -invariant irreducible constituent α ∈ Irr( M ) with degree not divisible by p .Clearly, α is stabilized by M S , and | M S : S | is p -power and so is relativelyprime to α (1). Furthermore, the determinantal order o ( α ) is not divisible by p since M = O p ( M ). and thus o ( α ) is also relatively prime to | M S : S | . Itfollows from [4, Corollary 6.28] that α extends to some irreducible character β ∈ Irr(
M S ).Next, β G (1) = β (1) | G : M S | = α (1) | G : M S | is not divisible by p , and henceit has some constituent χ ∈ Irr( G ) with degree not divisible by p . Assume thatcod( χ ) = | G : ker χ | /χ (1) is not divisible by p . Then we deduce that S ≤ ker χ ,in particular, P ≤ ker χ . Thus, the irreducible constituents of χ P are trivial.By Frobenius Reciprocity, β is an irreducible constituent of χ MS and so α is an2rreducible constituent of χ M . Similarly, λ is an irreducible constituent of α P .Thus, λ is an irreducible constituent of χ P and is trivial, a contradiction.Assume that cod( χ ) is divisible by p . By hypothesis, χ is not a memberof Irr( G | N ′ ) and thus N ′ ≤ ker χ , and so the irreducible constituents of χ N are linear. The irreducible constituents of χ M are therefore linear, and in par-ticular, α is linear. Thus α is an extension of λ to M , and it follows that o ( λ ) divides o ( α ), which is not divisible by p , and therefore o ( λ ) is not divisibleby p . This is a contradiction since λ is a nontrivial linear character of a p -group.Now, we prove that N is solvable. By the above arguments, we know that N has a normal p -complement K , and we write M = N ′ ∩ K . Then p doesnot divide | M | and every member of cod( G | M ) is not divisible by p . Let S ∈ Syl p ( G ) and note that if θ ∈ Irr( M ) is S -invariant, then θ is extendible to somecharacter ϕ ∈ Irr(
M S ). Since p does not divide ϕ G (1) = | G : M S | ϕ (1), we seethat ϕ G has an irreducible constituent χ with degree not divisible by p .Assume that cod( χ ) = | G : ker χ | /χ (1) is not divisible by p . Then S ≤ ker χ and thus S ≤ ker ϕ . We deduce that ϕ ( ms ) = ϕ ( m ) for all m ∈ M and s ∈ S .So every θ ∈ Irr( M ) is S -invariant. It follows from Glauberman’s theorem that C M ( S ) = M . Thus M S = M × S and ϕ = θ × S . Since S = 1, we may chooseagain ϕ = θ × ξ ∈ Irr(
M S ) for some nonprincipal character ξ ∈ Irr( S ). So S (cid:2) ker ϕ , which is a contradiction.Thus cod( χ ) is divisible by p . It follows that χ cod( G | N ′ ) and hence M ≤ ker χ and θ = 1 M . It follows again from Glauberman’s theorem that C M ( S ) = 1. By Lemma 2.1, M is solvable, and thus N is solvable, as claimed. (cid:3) Acknowledgements
J. Lu is supported by National Natural Science Foundation of China (1186-1015), Guangxi Natural Science Foundation Program (2020GXNSFAA238045),and Training Program for 1,000 Young and Middle-aged Cadre Teachers inUniversities of Guangxi Province; B. Zhang is supported by Guangxi BasicAbility Promotion Project for Young and Middle-aged Teachers (2020KY02019).
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