Automorphism property space

This is an abstraction from a general concept of properties.

An automorphism property is a map from the collection of all group automorphisms to the two-element set (true, false), with the property that if there is an isomorphism identifying two group automorphisms, they either both have the property or both do not have the property.

The automorphism property space is the collection of all automorphism properties.

Elements in the automorphism property space

The tautology and fallacy

The space of automorphism properties is naturally equipped with a Boolean lattice structure. The two extreme elements of this lattice are the tautology -- the property satisfed by all automorphisms, and the fallacy -- the property satisfied by no automorphism.

Subgroup-defining properties

An automorphism property is termed subgroup-defining if, for any group, the set of automorphisms of the group satisfying the property forms a subgroup of the automorphism group of that group. The tautology, for instance, is a subgroup-defining property.

Some important elements in the automorphism property space

Some of the automorphism properties we frequently encounter:

• The property of being an inner automorphism
• The property of being a class automorphism
• The property of being a family automorphism
• The property of being an extensible automorphism

All these are subgroup-defining.

A more complete listing of important automorphism properties is available at:

Category:Automorphism properties

Property modifiers/operators

Extensibility map

In its full generality, the extensibility map takes an automorphism property $\alpha$, a subgroup property $p$, and returns the property $\beta$ such that:

for a group $G$, an automorphism $\sigma$ satisfies $\beta$ if, given any embedding $G \leq H$ satisfying property $p$, there is an automorphism $\phi$ of $H$ such that $\phi$ satisfies $\alpha$ and the restriction of $\phi$ to $H$ is $\sigma$.

When $\alpha$ is taken to be the tautology, we get a map from the automorphism property space to itself, which is what we usually mean when we talk of the extensibility map.

The extensibility map is a monotone ascendant map. We can thus apply iteration operators to it. Some standard terminology: