en>Mark viking: gate -> logic gate, added wl

2014-01-06T09:57:06Z

gate -> logic gate, added wl

← Older revision		Revision as of 11:57, 6 January 2014
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	~~29 yr old Aboriginal and Torres Strait Islander Health Worker Paris~~ from ~~Vermilion River County No~~. 24, ~~likes water skiing~~, ~~underwear size~~ and ~~urban exploration~~. Is a ~~travel maniac and recently traveled~~ to ~~Agra Fort~~.<br><br>~~Look into my web-site~~ :: [http://~~www~~.~~sociallinks~~.~~info~~/~~story~~.~~php?title=how-much-you-need-to-expect-youll-pay-for-a-good-underwear-size underwear sizes chart~~]		In [[group theory]], a branch of [[mathematics]], an '''opposite group''' is a way to construct a [[group (mathematics)\|group]] from another group that allows one to define [[Group action\|right action]] as a special case of left action.

			== Definition ==
			Let <math>G</math> be a group under the operation <math></math>. The opposite group of <math>G</math>, denoted <math>G^{op}</math>, has the same underlying set as <math>G</math>, and its group operation <math>\mathbin{\ast'}</math> is defined by <math>g_1 \mathbin{\ast'} g_2 = g_2 g_1</math>.

			If <math>G</math> is [[abelian group\|abelian]], then it is equal to its opposite group. Also, every group <math>G</math> (not necessarily abelian) is [[naturally isomorphic]] to its opposite group: An isomorphism <math>\varphi: G \to G^{op}</math> is given by <math>\varphi(x) = x^{-1}</math>. More generally, any anti-automorphism <math>\psi: G \to G</math> gives rise to a corresponding isomorphism <math>\psi': G \to G^{op}</math> via <math>\psi'(g)=\psi(g)</math>, since
			: <math>\psi'(g * h) = \psi(g * h) = \psi(h) * \psi(g) = \psi(g) \mathbin{\ast'} \psi(h)=\psi'(g) \mathbin{\ast'} \psi'(h).</math>

			== Group action ==
			Let <math>X</math> be an object in some category, and <math>\rho: G \to \mathrm{Aut}(X)</math> be a [[Group action\|right action]]. Then <math>\rho^{op}: G^{op} \to \mathrm{Aut}(X)</math> is a left action defined by <math>\rho^{op}(g)x = \rho(g)x</math>, or <math>g^{op}x = xg</math>.

			== External links ==
			* [http://planetmath.org/encyclopedia/OppositeGroup.html http://planetmath.org/encyclopedia/OppositeGroup.html]


			[[Category:Group theory]]
			[[Category:Representation theory]]

24.118.123.34 at 03:29, 2 April 2011

2011-04-02T03:29:18Z

New page

29 yr old Aboriginal and Torres Strait Islander Health Worker Paris from Vermilion River County No. 24, likes water skiing, underwear size and urban exploration. Is a travel maniac and recently traveled to Agra Fort.<br><br>Look into my web-site :: [http://www.sociallinks.info/story.php?title=how-much-you-need-to-expect-youll-pay-for-a-good-underwear-size underwear sizes chart]

Switching lemma - Revision history

en>Mark viking: gate -> logic gate, added wl

24.118.123.34 at 03:29, 2 April 2011