The Finite Simple Groups

The finite simple groups are the building blocks from which all the finite groups are made and as such are objects of fundamental importance throughout mathematics. They can be divided into the alternating, sporadic and Lie type groups, the latter further subdivided into classical and exceptional, or into untwisted and twisted types. This book is the first accessible introduction to all these families of finite simple groups at a level suitable for final year undergraduate and beginning graduate students. The first five chapters provide a thorough grounding in the theory of the alternating and classical groups, followed by an overview of the exceptional groups (treated as automorphism groups of multilinear forms) and the sporadic groups. These chapters form the basis of a final year undergraduate course bringing their undergraduate studies to a fitting climax with seminal results from the late 20th century. The final two chapters give an introduction to the theory of Lie algebras and Chevalley groups (which provides a unified approach to all the untwisted finite groups of Lie type) and to algebraic groups (which unites the twisted and untwisted types). These final chapters are ideal guides for undergraduate projects and prepare the students for further reading in more advanced texts on these important topics.