Groupprops, The Group Properties Wiki (pre-alpha)
YOUR FEEDBACK IS IMPORTANT!
Please take a short user satisfaction survey about Groupprops.
Your survey responses will be helpful in improving the site experience!
Thanks in advance!
Variety-extensible automorphism
From Groupprops
|
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a property that can be evaluated for an automorphism of an algebra in a variety of algebras. The evaluation of that property depends on the ambient variety, and not just on the automorphism or the algebra.
View all such properties
Definition
Let 
be a variety of algebras, and A be an algebra in . An automorphism σ of A is termed extensible' over the variety if, whenever A is embedded as a subalgebra of an algebra B of , there exists an automorphism 
of B such that the restriction of to A is σ.
Particular cases
Variety of sets
In the variety of sets, every automorphism is extensible. In other words, given any set and a subset, a permutation of the subset always extends to a permutation of the whole set.
The idea here is that adding more elements to a set does not destroy the inherent symmetry between the elements already there.
Variety of groups
Further information: extensible automorphism
In the variety of groups, every inner automorphism is extensible. The extensible automorphisms conjecture states that every extensible automorphism is inner.
Relation with other properties
Stronger properties
- I-automorphism
- Variety-infinity-extensible automorphism
- Variety-pushforwardable automorphism
- Variety-infinity-pushforwardable automorphism
