On a Certain Lie Algebra Defined by a Finite Group

1. INTRODUCTION. Some years ago W. Plesken told the first author of a simple but interesting construction of a Lie algebra from a finite group. The authors posed themselves the question as to what the structure of this Lie algebra might be. In particular, for which groups does the construction produce a simple Lie algebra? The answer is given in the present paper; it uses some textbook results on representations of finite groups, which we explain along the way. Little knowledge of the theory of Lie algebras is required beyond the definition of a Lie algebra itself and the definitions of simple and semisimple Lie algebras. Thus this exposition may serve as the basis for some entertaining examples or exercises in a graduate course on the representation theory of finite groups.