Difference between revisions of "Semi-Direct Products"
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| − | 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 \ | + | 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>. |
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| + | '''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 </math>, where <math>j^n\equiv 1 \mod m </math>. | ||
Revision as of 23:12, 11 December 2022
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 \( C_m \rtimes C_N \cong \langle r,s \mid r^m = e, s^n = e, srs^{-1} = r^j \), where \(j^n\equiv 1 \mod m \).
