Difference between revisions of "Semi-Direct Products"
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Kfagerstrom (talk | contribs) (Created page with " Let <math>H </math> and <math> K</math> be groups and let <math>\rho:K\to \Aut(H) </math> be a homomorphism. The (external) semidirect product induced by <math>\rho </math> i...") |
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| − | Let <math>H </math> and <math> K</math> be groups and let <math>\rho:K\to \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_p K</math> |
Revision as of 23:35, 7 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 \( H \rtimes_p K\)
