Semi-Direct Products - Revision history

Kfagerstrom at 23:16, 11 December 2022

2022-12-11T23:16:19Z

Kfagerstrom at 23:14, 11 December 2022

2022-12-11T23:14:59Z

Kfagerstrom at 23:12, 11 December 2022

2022-12-11T23:12:00Z

Kfagerstrom at 23:35, 7 December 2022

2022-12-07T23:35:44Z

Kfagerstrom: Created page with " Let $H$ and $K$ be groups and let $\rho:K\to \Aut(H)$ be a homomorphism. The (external) semidirect product induced by $\rho$ i..."

2022-12-07T23:34:52Z

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 \coprod_p 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>.	+	'''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>.

← Older revision		Revision as of 23:14, 11 December 2022
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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>.	+	'''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>.

← Older revision		Revision as of 23:12, 11 December 2022
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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 \~~rtimes_p~~ K</math>	+	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>.