Normal P-complement Research Articles

Primes and degrees of Isaacs π-partial characters

Let G be a finite π -solvable group. In this paper, we show that all monomial irreducible Isaacs π -partial character degrees of G are π -number, if and only if G has a normal Hall π ′ -subgroup. Furthermore, let G be a finite π -separable group and let p be prime in π . We prove that if p divides φ ( 1 ) for all non-linear φ ∈ I π ( G ) , then G has a normal p-complement.

Conjugacy classes and union of cosets of normal subgroups

Let G be a finite group, N a normal subgroup of G and K a conjugacy class of G. We prove that if Kcup K^{-1} is union of cosets of N, then N is soluble, K is a real-imaginary class, that is, every irreducible character of G takes real or purely imaginary values on K, and if, in addition, the elements of K are p-elements for some prime p, then N has normal p-complement. We also prove that if Kcup K^{-1} is a single coset of N, then langle Krangle has a normal 2-complement.

On existence of normal p-complement of finite groups with restrictions on the conjugacy class sizes

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that $Ind_G(a)_p\in \{1,p^{\alpha}\}$ for every $a$ of $G$ and a $p$-element $x\in G$ such that $Ind_G(x)_p>1$, then $G$ includes a normal $p$-complement.

Kernels of $$p'$$-Degree Irreducible Characters

Let G be a finite group and let p be a prime number. We prove that if \(\chi \in {{{\text {Irr}}}}_{p'}(G)\) and \({\text {Ker}}\chi \) does not have a solvable normal p-complement then there exists \(\psi \in {{{\text {Irr}}}}_{p'}(G)\) such that \(\psi (1)>\chi (1)\) and \({\text {Ker}}\psi <{\text {Ker}}\chi \). This is a \(p'\)-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees.

Notes on normal p-complements and monomial characters

Let G be a finite group 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) as the codegree of χ, where ker χ is the kernel of χ. Denote by the intersection of the kernels of all the monolithic, monomial irreducible characters of a p-solvable group G with codegrees divisible by a prime p. We prove in this note that has a normal p-complement, and then the theorem in [2] is generalized. Also, we prove that about the theorem in [2] more is true.

MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

AbstractLet G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

English

Let G be a finite group. We prove that if every self-centralizing subgroup of G is nilpotent or subnormal or a TI-subgroup, then every subgroup of G is nilpotent or subnormal. Moreover, G has either a normal Sylow p-subgroup or a normal p-complement for each prime divisor p of |G|.

Normal p-complements and monomial characters

Let G be a finite group, p be a prime and $$\mathrm{Irr}(G)$$ be the set of irreducible (complex) characters of G. Let $$\chi \in \mathrm{Irr}(G)$$ and write $$\mathrm{cod}(\chi )=|G: \ker \chi |/\chi (1)$$, where $$\ker \chi $$ is the kernel of $$\chi $$. We prove in this note that if G is solvable and $$\mathrm{cod}(\chi )$$ is a $$p'$$-number for every monomial character $$\chi \in \mathrm{Irr}(G)$$, then G has a normal p-complement.

Finite groups satisfying character degree congruences

Abstract Let G be a finite group, p a prime and c ∈ { 0 , 1 , … , p - 1 } {c\in\{0,1,\ldots,p-1\}} . Suppose that the degree of every nonlinear irreducible character of G is congruent to c modulo p. If here c = 0 {c=0} , then G has a normal p-complement by a well known theorem of Thompson. We prove that in the cases where c ≠ 0 {c\neq 0} the group G is solvable with a normal abelian Sylow p-subgroup. If p ≠ 3 {p\neq 3} then this is true provided these character degrees are congruent to c or to - c {-c} modulo p.

P-divisibility of conjugacy class sizes and normal p-complements

Abstract Let N be a normal subgroup of a group G and let p be a prime. We prove that if the p-part of |x| G is a constant for every prime-power order element x ∈ N∖Z(N), then N is solvable and has normal p-complement.

Finite groups whose noncentral class sizes have the same p-part for some prime p

We describe the structure of the finite groups in which the sizes of noncentral conjugacy classes have the same p-part, for some prime p. We prove that they are solvable, they have a normal p-complement and their Fitting length is at most three.

Remarks on chains of weakly closed subgroups

Abstract In this paper we generalize and unify several results proved in recent papers about the existence of normal p-complements and conditions on supersolubility. Moreover a counterexample is given to a question in [Guo and Wei, J. Group Theory 13: 267–276, 2010] and it is proved that a finite group is 2-nilpotent if the cyclic subgroups of order less or equal than four are strongly closed.

Normal Restriction in Finite Groups

A subgroup H of a finite group G is called an NR − subgroup (Normal Restriction) if whenever K ⊴ H, then K G ∩ H = K, where K G is the normal closure of K in G. In this article, we will prove some sufficient conditions for the solvability of finite groups which possess many NR-subgroups. We also prove a criterion for the existence of a normal p-complement in finite groups.

Normal p-complements and fixed elements

We extend known results by proving that if an odd-order group acts on a 2-group and fixes all elements of order 2 and all real elements of order 4, then the action is trivial. We apply this to establish some 2-nilpotence criteria.

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element $${g\in G}$$ is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.

Degrees of rational characters of finite groups

Degrees of rational characters of finite groups

Large orbits of elements centralized by a Sylow subgroup

If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist $${v_1, v_2 \in V}$$ such that $${{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P}$$ , where $${P \in {\rm Syl}_p(G)}$$ . We hence deduce that, if the normal p-complement K is nontrivial, there exists $${v \in {\bf C}_{V}(P)}$$ such that |K : C K (v)|2 > |K|.

The structure of finite groups with three class sizes

Let G be a finite group and suppose that the set of conjugacy class sizes of G is f1; m; mng, where m; n > 1 are coprime. We prove that m ¼ p for some prime p dividing n � 1. We also show that G has an abelian normal p-complement and that if P is a Sylow p-subgroup of G, then jP 0 j¼ p and jP=ZðGÞp j¼ p 2 . We obtain other properties and deter- mine completely the structure of G.

Schur indices of characters in blocks of full defect

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that

Normal π-complements for finite groups

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and NG(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.