Published online by Cambridge University Press: 18 May 2009
Suppose that a group G is the semidirect product of a subgroup N and a normal subgroup M. Then the elements of G have unique expressions mn (m ∈ M, n ∈ N) and the commutator function
maps N x M into M. In fact there is an action (by automorphisms) of N on M given by
Conversely, if one is given an action of a group N on a group M then one can construct a semidirect product.