Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Definition 2 Particular cases 2.1 Variety of sets 2.2 Variety of groups 3 Relation with other properties 3.1 Stronger properties

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW:
VIEW RELATED:
Learn more about terminology local to the wiki | view a complete list of such terminology

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