Automorphism group of a group

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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 G, denoted \operatorname{Aut}(G), is a set whose elements are automorphisms \sigma:G \to G, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of \operatorname{Sym}(G), the group of all permutations on G.

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 g \in G such that the map has the form x \mapsto gxg^{-1} it is called the inner automorphism group and is isomorphic to the quotient group G/Z(G) where Z(G) 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

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 n, the automorphism group is of order \varphi(n) where \varphi denotes the Euler totient function. Further, the automorphism group is cyclic iff n is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime p, the automorphism group of the cyclic group of order p is the cyclic group of order p - 1.
cyclic group:Z3 3 cyclic group:Z2 2 endomorphism structure of cyclic group:Z3 For a finite cyclic group of order n, the automorphism group is of order \varphi(n) where \varphi denotes the Euler totient function. Further, the automorphism group is cyclic iff n is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime p, the automorphism group of the cyclic group of order p is the cyclic group of order p - 1.
cyclic group:Z4 4 cyclic group:Z2 2 endomorphism structure of cyclic group:Z4 For a finite cyclic group of order n, the automorphism group is of order \varphi(n) where \varphi denotes the Euler totient function. Further, the automorphism group is cyclic iff n is 2,4, a power of an odd prime, or twice a power of an odd prime.
Klein four-group 4 symmetric group:S3 6 endomorphism structure of Klein four-group In general, for an elementary abelian group of order p^n, the automorphism group is the general linear group GL(n,p).
cyclic group:Z5 5 cyclic group:Z4 4 endomorphism structure of cyclic group:Z5 For a finite cyclic group of order n, the automorphism group is of order \varphi(n) where \varphi denotes the Euler totient function. Further, the automorphism group is cyclic iff n is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime p, the automorphism group of the cyclic group of order p is the cyclic group of order p - 1.
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 n, the automorphism group is of order \varphi(n) where \varphi denotes the Euler totient function. Further, the automorphism group is cyclic iff n is 2,4, a power of an odd prime, or twice a power of an odd prime.
cyclic group:Z7 7 cyclic group:Z6 6 endomorphism structure of cyclic group:Z7 In particular, for a prime p, the automorphism group of the cyclic group of order p is the cyclic group of order p - 1.
cyclic group:Z8 8 Klein four-group 4 endomorphism structure of cyclic group:Z8 For a finite cyclic group of order n, the automorphism group is of order \varphi(n) where \varphi denotes the Euler totient function. Further, the automorphism group is cyclic iff n is 2,4, a power of an odd prime, or twice a power of an odd prime.
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 p^n, the automorphism group is the general linear group GL(n,p). In this case, n = 3, p = 2, so we get GL(3,2), which is isomorphic to PSL(3,2).
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 S_n is a complete group if n \ne 2,6.

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 n, it is an abelian group of order \varphi(n) defined as the multiplicative group modulo n. It is itself cyclic if n = 2,4, a power of an odd prime, or twice a power of an odd primes 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:

Order Information on group cohomology
8 Endomorphism structure of groups of order 8
12 Endomorphism structure of groups of order 12
16 Endomorphism structure of groups of order 16
18 Endomorphism structure of groups of order 18
20 Endomorphism structure of groups of order 20
24 Endomorphism structure of groups of order 24
48 Endomorphism structure of groups of order 48