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

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 cyclic group:Z4 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.