Finite simple groups: introduction and examples

In this thesis, we study finite simple groups as well as the Classification Theorem, which classifies all such groups. We will see by the Classification Theorem that all simple groups are one of the following four types: cyclic of prime order, alternating on n elements for n 2: 5, of Lie-type, or sporadic. We prove that the cyclic groups of prime order are the only finite simple Abelian groups, and that the alternating groups on n elements are indeed simple for n 2: 5. We also prove that all groups in a specific family of Lie-type groups, namely PSL2 (q), where q is a power of a prime and q > 3, are simple. Even though all finite simple groups have currently been classified, we explore some methods used to show certain groups are not simple. Through this, we are able to classify all simple groups of order through 360, as well as provide some more general results that cover infinitely many groups. Keywords: groups, simple groups, Sylow Theorems, Jordan-Holder Theorem, Classification Theorem