Automorphism of a group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a function property, viz a property of functions from a group to itself

Automorphism redirects here. For more general notions of automorphism, refer automorphism of a universal algebra and automorphism of a structure

Definition

Symbol-free definition

An automorphism of a group is any of the following equivalent things:

• An isomorphism from the group to itself
• A bijective endomorphism of the group
• A homomorphism that is both an endomorphism and an isomorphism

Definition with symbols

Let $G$ be a group. A map $\sigma$ from $G$ to itself is termed an automorphism of $G$ if it satisfies all of the following conditions:

• $\sigma$ is bijective
• $\sigma(gh) = \sigma(g)\sigma(h)$ whenever $g$ and $h$ are both in $G$
• $\sigma(e) = e$
• $\sigma(g^{-1}) = \sigma(g)^{-1}$

Actually, the third and fourth condition follow from the first two (refer equivalence of definitions of group homomorphism).

Relation with other properties

Stronger properties

These are properties of (function from a group to itself) that are stronger than the property of being an automorphism:

• Inner automorphism: An automorphism that arises as conjugation via an element of the group
• Power automorphism: An automorphism that maps each element to a power of itself

Weaker properties

These are properties of (function from a group to itself) that that are weaker than the property of being an automorphism, viz function properties that are satisfied by every automorphism:

• Injective endomorphism
• Surjective endomorphism
• Endomorphism
• Semiautomorphism
• Quasiautomorphism

Facts

Composition

The composite of two automorphisms of a group is again an automorphism of the group. This follows from the fact that the composite of any two isomorphisms is an isomorphism.

Inverse

The inverse of any automorphism is an automorphism.

Identity map

The identity map is always an automorphism.

Group structure

Combining the fact that automorphisms are closed under composition, inverse and contain the identity map, the automorphisms of a group form a subgroup of the monoid of all fucntions from the group to itself. This subgroup is termed the automorphism group of the given group.

Property theory

Inner automorphisms

Further information: inner automorphism

There is a natural homomorphism from any group to its automorphism group, that sends each element of the group to the conjugation map by that element. The image of the group under this map is termed the inner automorphism group, and automorphisms arising as such images are termed inner automorphisms.

Related subgroup properties

Characteristic subgroup

Further information: characteristic subgroup

A subgroup of a group is said to be characteristic if every automorphism of the group maps it to itself.

Characteristic subgroups are thus the automorphism-invariant subgroups (the invariance property for automorphisms)and the property of being characteristic is stronger than the property of being normal, which is just invariance under inner automorphisms. Characteristic subgroups also arise naturally as images of subgroup-defining functions.

Automorphs

Two subgroups of a group are said to be automorphs if there is an automorphism of the whole group that maps one subgroup to the other. Clearly, for a characteristic subgroup, it has no automorph other than itself. Subgroups which are automorphs are equivalent in all respects.

Conjugate subgroups are automatically automorphs, because every inner automorphism is also an automorphism.