scholarly journals On order prime divisor graphs of finite groups

2021 ◽  
Vol 41 (2) ◽  
pp. 419
Author(s):  
Sumanta Das ◽  
Sunil K. Maity ◽  
Mridul K. Sen
Keyword(s):  
Finite Groups ◽  
Prime Divisor

scholarly journals ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS

2011 ◽  
Vol 53 (2) ◽  
pp. 401-410 ◽  
Author(s):  
LONG MIAO
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Divisor ◽  
Proper Subgroup ◽  
A Finite Group

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.

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
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].

scholarly journals CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

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.

scholarly journals On Sylow intersections

1977 ◽  
Vol 16 (2) ◽  
pp. 237-246 ◽  
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.

scholarly journals A note onp-solvable and solvable finite groups

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.

Enumeration of Finite Groups

2007 ◽  
Author(s):  
Simon R. Blackburn ◽  
Peter M. Neumann ◽  
Geetha Venkataraman
Keyword(s):  
Finite Groups
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