On a certain Lie algebra defined by a finite group

Arjeh M. Cohen and D. E. Taylor

Abstract

Following W. Plesken, let G be a finite group and let L(G) denote the Lie subalgebra of the complex group algebra of G spanned by the elements g - g^-1 where g belongs to G. The Lie product is [a,b] = ab-ba. In this paper we determine the structure of L(G). In particular, with two exceptions, L(G) is simple if and only if G is an extraspecial 2-group.

Keywords: finite group, Lie algebra, Schur-Frobenius indicator.

AMS Subject Classification: Primary 20.