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Automorphism group of a group

, 00:51, 25 February 2014
Particular groups
===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. Examples Some of the most important examples are given below: {| class="sortable" border="1"! 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===  {| class="sortable" border="1"! 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 being 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 || [[inner 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 automorphismgroup is the [[general linear group]]$GL(n, p)$. In this case, $n = 3, p = 2$, so we get $GL(3,2)$, which [[isomorphism between linear groups over field:F2|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: {| class ="sortable" border="1"! 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 [[automorphismgroup 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., [[extensible alternating group:A6]]), the group [[automorphismgroup 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: {| class="sortable" border="1"! 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]]|}