View source for Semi-Direct Products


Let <math>H </math> and <math> K</math> be groups and let <math>\rho:K\to \text{Aut}(H) </math> be a homomorphism. The (external) semidirect product induced by <math>\rho </math> is the set <math>H\times K </math> with the binary operation defined by <math>(h,k)(h',k')=(h\rho(k)(h') ,kk')</math>. This group is denoted by <math> H \rtimes_\rho K</math>.


'''Example'''. The semi-direct product of two cyclic groups, <math>C_m</math> and <math>C_n </math> has presentation <math> C_m \rtimes C_n \cong \langle r,s \mid r^m = e, s^n = e, srs^{-1} = r^j \rangle </math>, where <math>j^n\equiv 1 \mod m </math>.