Do Normal Subgroups Have Straight Tails?

1. INTRODUCTION. In this article we show that group representations permit simple, visual explanations of semidirect products and related concepts. Along the way, we prove an extension of Cayley's theorem, define blocks and tails for the resulting representations, and investigate the title question: Do normal subgroups have straight tails? We consider representations of finite groups over the field of complex numbers. Specifically, we define a representation of degree n of a group G to be a homomor-phism from G into a multiplicative group of complex n × n matrices. A more general viewpoint is certainly possible. If F is any field, then an F-representation of G is a homomorphism from G into the automorphism group of some F-vector space. Our first example is somewhat trivial. View the cyclic group of order three as the powers (modulo 3) of a single variable x: Z 3 = =x : x