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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Definition
Symbol-free definition
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
.
Subgroups
Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Some of the most important examples are given below:
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 ![]() ![]() |
it is called the inner automorphism group and is isomorphic to the quotient group ![]() ![]() |
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 |
Facts
- 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.
Particular cases
Particular groups
Group | Order | Automorphism group | Order | Endomorphism structure page | More information |
---|---|---|---|---|---|
trivial group | 1 | trivial group | 1 | ||
cyclic group:Z2 | 2 | trivial group | 1 | endomorphism structure of cyclic group:Z2 | For a finite cyclic group of order ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
cyclic group:Z3 | 3 | cyclic group:Z2 | 2 | endomorphism structure of cyclic group:Z3 | For a finite cyclic group of order ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
cyclic group:Z4 | 4 | cyclic group:Z2 | 2 | endomorphism structure of cyclic group:Z4 | For a finite cyclic group of order ![]() ![]() ![]() ![]() |
Klein four-group | 4 | symmetric group:S3 | 6 | endomorphism structure of Klein four-group | In general, for an elementary abelian group of order ![]() ![]() |
cyclic group:Z5 | 5 | cyclic group:Z4 | 4 | endomorphism structure of cyclic group:Z5 | For a finite cyclic group of order ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
symmetric group:S3 | 6 | symmetric group:S3 | 6 | endomorphism structure of symmetric group:S3 | See symmetric groups are complete and endomorphism structure of symmetric group:S3 |
cyclic group:Z6 | 6 | cyclic group:Z2 | 2 | endomorphism structure of cyclic group:Z6 | For a finite cyclic group of order ![]() ![]() ![]() ![]() |
cyclic group:Z7 | 7 | cyclic group:Z6 | 6 | endomorphism structure of cyclic group:Z7 | In particular, for a prime ![]() ![]() ![]() |
cyclic group:Z8 | 8 | Klein four-group | 4 | endomorphism structure of cyclic group:Z8 | For a finite cyclic group of order ![]() ![]() ![]() ![]() |
direct product of Z4 and Z2 | 8 | dihedral group:D8 | 8 | endomorphism structure of direct product of Z4 and Z2 | |
dihedral group:D8 | 8 | dihedral group:D8 | 8 | endomorphism structure of dihedral group:D8 | |
quaternion group | 8 | symmetric group:S4 | 24 | endomorphism structure of quaternion group | |
elementary abelian group:E8 | 8 | projective special linear group:PSL(3,2) | 168 | endomorphism structure of projective special linear group:PSL(3,2) | In general, for an elementary abelian group of order ![]() ![]() ![]() ![]() ![]() |
alternating group:A4 | 12 | symmetric group:S4 | 24 | endomorphism structure of alternating group:A4 | |
symmetric group:S4 | 24 | symmetric group:S4 | 24 | endomorphism structure of symmetric group:S4 | symmetric groups are complete: the symmetric group ![]() ![]() |
Group families
For various group families, the automorphism group can be described in terms of parameters for members of the families. The descriptions are sometimes quite complicated, so we simply provide links:
Family | Description of automorphism group | Endomorphism structure information |
---|---|---|
finite cyclic group | For a cyclic group of order ![]() ![]() ![]() |
endomorphism structure of finite cyclic groups |
finite abelian group | (no simple description) | -- |
symmetric group | the same symmetric group if the degree is not 2 or 6. For degree 2, the trivial group. For degree 6 (i.e., symmetric group:S6), the group automorphism group of alternating group:A6. | endomorphism structure of symmetric groups |
alternating group | the symmetric group if the degree is at least 3 and not equal to 6. For degree 6 (i.e., alternating group:A6), the group automorphism group of alternating group:A6. | endomorphism structure of alternating groups |
Grouping by order
We give below the information for the group cohomology (and hence in particular, the Schur multipliers) for groups of small orders: