Semi-Direct Products

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

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