Automorphism group of a group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Automorphism group of a group, all facts related to Automorphism group of a group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]

View a complete list of basic definitions in group theory | Go through a guided tour for beginners to this wiki

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 $A u t (G)$ , is a set whose elements are automorphisms $σ : G \to G$ , and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of $S y m (G)$ , 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 $x \mapsto g x g^{- 1}$	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 $φ (n)$ where $φ$ 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 $φ (n)$ where $φ$ 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 $φ (n)$ where $φ$ 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 $G L (n, p)$ .
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 $φ (n)$ where $φ$ 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 $φ (n)$ where $φ$ 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 $φ (n)$ where $φ$ 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 $G L (n, p)$ . In this case, $n = 3, p = 2$ , so we get $G L (3, 2)$ , which is isomorphic to $P S L (3, 2)$ .
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 $n \neq 2, 6$ .