A Study About One Generation of Finite Simple Groups and Finite Groups

Nader Taffach

doi:10.5539/jmr.v13n3p59

Abstract

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1&nbsp; and p_2&nbsp; are two different primes. We also show that for a given different prime numbers p&nbsp; and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

Highlights

Finite groups often arise when considering the symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations
In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1- and p2-subgroups, where p1 and p2 are two different primes
We show that for a given different prime numbers p and q, any finite group can be generated by a Sylow p-subgroup and a q-subgroup

Summary

Introduction

Finite groups often arise when considering the symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. In (Aschbacher & Guralnick, 1984) authors showed that any sporadic simple group can be generated by an involution and another suitable element, recall that a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups (Wilson, 1998). In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1- and p2-subgroups, where p1 and p2 are two different primes.

Generations of Simple Groups

Generations of Finite Groups