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
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 be a group. A map from to itself is termed an automorphism of if it satisfies all of the following conditions:
- is bijective
- whenever and are both in
Actually, the third and fourth condition follow from the first two (refer equivalence of definitions of group homomorphism).
Relation with other 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
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:
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.
The inverse of any automorphism is an automorphism.
The identity map is always an automorphism.
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.
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
Further information: characteristic subgroup
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.
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.