Automorphism group 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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The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.
Definition with symbols
The automorphism group of a group , denoted , is a set whose elements are automorphisms , and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of , the group of all permutations on .
|Group-closed automorphism property||Meaning||Corresponding normal subgroup of the automorphism group|
|inner automorphism||can be expressed as conjugation by an element of the group, i.e., there exists such that the map has the form||it is called the inner automorphism group and is isomorphic to the quotient group where is the center. See group acts as automorphisms by conjugation.|
|class-preserving automorphism||sends every element to within its automorphism class||the class-preserving automorphism group|
|IA-automorphism||sends every coset of the derived subgroup to itself, or equivalently, induces the identity map on the abelianization.||the IA-automorphism group|
|center-fixing automorphism||fixes every element of the center||the center-fixing automorphism group|
|monomial automorphism||can be expressed using a monomial formula||the momomial automorphism group|
|normal automorphism||sends every normal subgroup to itself||the normal automorphism group|
- Extensible equals inner: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an inner automorphism.
- Quotient-pullbackable equals inner: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.