On order prime divisor graphs of finite groups

AbstractA subgroup H is called weakly -supplemented in a finite group G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. In this paper we will prove the following: Let G be a finite group and P be a Sylow p-subgroup of G, where p is the smallest prime divisor of |G|. Suppose that P has a non-trivial proper subgroup D such that all subgroups E of P with order |D| and 2|D| (if P is a non-abelian 2-group, |P : D| > 2 and there exists D1 ⊴ E ≤ P with 2|D1| = |D| and E/D1 is cyclic of order 4) have p-nilpotent supplement or weak -supplement in G, then G is p-nilpotent.

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On the Frobenius spectrum of a finite group

Journal of Algebra and Its Applications ◽

10.1142/s0219498817500517 ◽

2017 ◽

Vol 16 (03) ◽

pp. 1750051 ◽

Cited By ~ 1

Author(s):

Jiangtao Shi ◽

Wei Meng ◽

Cui Zhang

Keyword(s):

Finite Group ◽

Positive Integer ◽

Finite Groups ◽

Prime Divisor ◽

Complete Classification ◽

Composition Factor ◽

Frobenius Theorem ◽

Even Order ◽

A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

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CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002735 ◽

2012 ◽

Vol 85 (2) ◽

pp. 217-231

Author(s):

CARLO CASOLO ◽

ELISA MARIA TOMBARI

Keyword(s):

Conjugacy Class ◽

Finite Groups ◽

Prime Divisor ◽

Full Description ◽

Direct Products ◽

Conjugacy Class Sizes

AbstractWe consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

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On Sylow intersections

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270002325x ◽

1977 ◽

Vol 16 (2) ◽

pp. 237-246 ◽

Cited By ~ 1

Author(s):

Ariel Ish-Shalom

Keyword(s):

Finite Group ◽

Finite Groups ◽

Prime Divisor ◽

A Finite Group

Let G be a finite group, p a prime divisor of |G|, and T a p–subgroup of G. Define σ(T) to be the number of Sylow p–subgroups of G containing T. Call T a central p–Sylow intersection if for some Σ ⊆ Sylp (G), T = ∩(S | S є Σ), and if, in addition, T contains the center of a Sylow p–subgroup of G. This work is inspired and motivated by work of G. Stroth [J. Algebra 37 (1975), 111–120]. Generalizing an argument of his we describe finite groups in which every central p–Sylow intersection T with p–rank(T) > 2 satisfies σ(T) ≤ p.Related methods yield the description of finite groups in which every central p–Sylow intersection T with p–rank(T) ≥ 2 satisfies σ(T) ≤ 2p.

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A note onp-solvable and solvable finite groups

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171294001158 ◽

1994 ◽

Vol 17 (4) ◽

pp. 821-824

Author(s):

R. Khazal ◽

N. P. Mukherjee

Keyword(s):

Finite Groups ◽

Prime Divisor ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient

The notion of normal index is utilized in proving necessary and sufficient conditions for a groupGto be respectively,p-solvable and solvable wherepis the largest prime divisor of|G|. These are used further in identifying the largest normalp-solvable and normal solvable subgroups, respectively, ofG.

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