Extensible automorphism
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2007-08-06T17:33:39Z
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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 <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>.
Every [[inner automorphism]] of a group is <math>\omega</math> extensible. The question of whether every extensible automorphism of a group is inner is an open problem. Here are the results obtained 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.
==External links==
* [http://www.cmi.ac.in/~vipul/unsolvedproblems/ Page giving info on extensible automorphisms]
[[Category:Group theory]]
[[Category:Group automorphisms]]