Extensible automorphism
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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 is an automorphism that can be lifted to an automorphism of any group in which it is embedded.
A k times extensible automorphism of a group is defined inductively as an automorphism that can be lifted to a k − 1 times extensible automorphism for any embedding, where a 0 times extensible automorphism is simply any automorphism. An automorphism that is k times extensible for all k is termed an ωextensible automorphism. The k extensible automorphisms of a group form a subgroup for every k.
Every inner automorphism of a group is ω 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 homomorphic image acting on another group such that the other group is 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.
