Search results for "Simple group"

On groups with abelian Sylow 2-subgroups

1970

Finite groups with abelian Sylow 2-subgroups have been classified by Walter [8]. In this note I want to describe an alternate proof of some partial result of Walter's work, namely the theorem stated below. It represents the first major reduction step in that classification. The approach used here is to some extent derived from [1]. ! Besides the groups L 2 (q)= PSL(2, q) another class of simple groups enters our discussion: We say that a simple group G with abelian Sz-subgroups is of type JR (Janko-Ree) if, for any involution t in G, CG (t) is a maximal subgroup of G isomorphic to ( t ) | E where PSL(2, q)~ E ~_ PFL(2, q) with odd q > 5. In fact, E = L 2 (q), as proved by Walter 1-7] ; and …

Injective Fitting sets in automorphism groups

1993

A Local Approach to Certain Classes of Finite Groups

2003

Abstract We develop several local approaches for the three classes of finite groups: T-groups (normality is a transitive relation) and PT-groups (permutability is a transitive relation) and PST-groups (S-permutability is a transitive relation). Here a subgroup of a finite group G is S-permutable if it permutes with all the Sylow subgroup of G.

Self-normalizing Sylow subgroups

2003

Using the classification of finite simple groups we prove the following statement: Let p > 3 p>3 be a prime, Q Q a group of automorphisms of p p -power order of a finite group G G , and P P a Q Q -invariant Sylow p p -subgroup of G G . If C N G ( P ) / P ( Q ) \mathbf {C}_{\mathbf {N}_G(P)/P}(Q) is trivial, then G G is solvable. An equivalent formulation is that if G G has a self-normalizing Sylow p p -subgroup with p > 3 p >3 a prime, then G G is solvable. We also investigate the possibilities when p = 3 p=3 .

Some results concerning simple locally finite groups of 1-type

2005

AbstractIn this paper several aspects of infinite simple locally finite groups of 1-type are considered. In the first part, the classes of diagonal limits of finite alternating groups, of diagonal limits of finite direct products of alternating groups, and of absolutely simple groups of 1-type are distinguished from each other. In the second part, inductive systems of representations over fields of characteristic zero (which are known to correspond to ideals in the group algebra) are studied in general for groups of 1-type. The roles of primitive respectively imprimitive representations in inductive systems are investigated. Moreover it is shown that in any proper inductive system the depth…

A reduction theorem for a conjecture on products of two π -decomposable groups

2013

[EN] For a set of primes pi, a group X is said to be pi-decomposable if X = X-pi x X-pi' is the direct product of a pi-subgroup X-pi and a pi'-subgroup X-pi', where pi' is the complementary of pi in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: "Let pi be a set of odd primes. If the finite group G = AB is a product of two pi-decomposable subgroups A = A(pi) x A(pi') and B = B-pi x B-pi', then A(pi)B(pi) = B(pi)A(pi) and this is a Hall pi-subgroup of G." We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. (C) 2013 Elsevier Inc. All ri…

Brauer characters and coprime action

2016

Abstract It is an open problem to show that under a coprime action, the number of invariant Brauer characters of a finite group is the number of the Brauer characters of the fixed point subgroup. We prove that this is true if the non-abelian simple groups satisfy a stronger condition.

A Uniform Way to Control Chief Series in Finite p -Groups and to Construct the Countable Algebraically Closed Locally Finite p -Groups

1986

On Finite Solvable Groups That Behave Like Nilpotent Groups with Respect to the Frattini Group

1994

Groups with few $p'$-character degrees

2019

Abstract We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group G contains only two distinct values not divisible by a given prime number p > 3 , then O p p ′ p p ′ ( G ) = 1 . This is done by using the classification of finite simple groups.