Multiple semidirect products of associative systems

Abstract

Suppose that a groupGis the semidirect product of a subgroupNand a normal subgroupM. Then the elements ofGhave unique expressionsmn(m∈M,n∈N) and the commutator functionmapsNxMintoM. In fact there is an action (by automorphisms) ofNonMgiven byConversely, if one is given an action of a groupNon a groupMthen one can construct a semidirect product.