en>David Eppstein: Remove orphan tag

2013-10-12T20:16:42Z

Remove orphan tag

← Older revision		Revision as of 22:16, 12 October 2013
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	~~Nice~~ to ~~meet you~~, ~~my name is Ling and I totally dig~~ that ~~name~~. ~~My home~~ is ~~now~~ in ~~Kansas~~. ~~The thing she adores most~~ is ~~to play handball but she can't make~~ it ~~her profession~~. ~~His day occupation~~ is a ~~financial officer but he plans on changing it.~~<br><br>~~Here~~ is ~~my web blog :: [http://megasport~~.bg/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId~~/~~73288/Default.aspx auto warranty~~]		In [[mathematics]], an [[automorphism]] of a structure is said to be '''extensible''' if, for any [[embedding]] of that structure inside another structure, the automorphism can be lifted to the bigger structure. In [[group theory]], an extensible automorphism of a [[group (mathematics)\|group]] is an automorphism that can be lifted to an automorphism of any group in which it is embedded. A group automorphism is extensible if and only if it is an [[inner automorphism]], {{harv\|Schupp\|1987}}.

			A <math>k</math> times extensible automorphism of a group is defined inductively as an automorphism that can be lifted to a <math>k-1</math> times extensible automorphism for any embedding, where a 0 times extensible automorphism is simply any automorphism. An automorphism that is <math>k</math> times extensible for all <math>k</math> is termed an <math>\omega</math> extensible automorphism. The <math>k</math> extensible automorphisms of a group form a [[subgroup]] for every <math>k</math>.

			Here are some properties in increasing order of generality:

			* The only extensible automorphism of an [[abelian group]] (extensible to arbitrary groups, not just to abelian groups) is the identity map.
			* Every extensible automorphism of a [[finite group]] is an [[IA automorphism]], that is, it acts as identity on the [[Abelianization]].
			* If a group has a [[homomorphism\|homomorphic]] image acting on another group such that the other group is [[characteristic subgroup\|characteristic]] in the [[semidirect product]] and the homomorphic image is a [[central factor]] in its [[normalizer]] in the semidirect product then any extensible automorphism of the group must get quotiented to an inner automorphism of its homomorphic image.

			==References==
			* {{Citation \| last1=Schupp \| first1=Paul E. \| title=A characterization of inner automorphisms \| doi=10.2307/2045986 \| mr=902532 \| year=1987 \| journal=Proceedings of the American Mathematical Society \| volume=101 \| issue=2 \| pages=226–228 \| publisher=American Mathematical Society \| jstor=2045986}}

			[[Category:Group theory]]
			[[Category:Group automorphisms]]

en>Brad7777: −Category:Projective geometry; ±Category:Theorems in geometry→Category:Theorems in projective geometry using HotCat

2012-02-05T16:28:40Z

−Category:Projective geometry; ±Category:Theorems in geometry→Category:Theorems in projective geometry using HotCat

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Nice to meet you, my name is Ling and I totally dig that name. My home is now in Kansas. The thing she adores most is to play handball but she can't make it her profession. His day occupation is a financial officer but he plans on changing it.<br><br>Here is my web blog :: [http://megasport.bg/ActivityFeed/MyProfile/tabid/60/userId/73288/Default.aspx auto warranty]

Intersection theorem - Revision history

en>David Eppstein: Remove orphan tag

en>Brad7777: −Category:Projective geometry; ±Category:Theorems in geometry→Category:Theorems in projective geometry using HotCat