A Study About One Generation of Finite Simple Groups and Finite Groups: دراسة حول أحد توليدات الزمر البسيطة المنتهية والزمر المنتهية

Abstract

The group theory and its classifications are of great importance in many engineering, physical and chemical fields, especially those related to the concept of symmetry. 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 P2-subgroups, where P1- and P2- are two different primes. We also, show that for a given different prime numbers  P and q, any finite group can be generated by a Sylow P- subgroup and an intravariant q- subgroup. The paper consists of an introduction and two fundamental sections. In one section we study the problem of generating simple finite groups. In another section, we mention the fundamental results of the paper, that connected with generating the finite group from some subgroups.