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→Particular groups

===Definition with symbols===

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

==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 <math>g \in G</math> such that the map has the form <math>x \mapsto gxg^{-1}</math> || it is called the [[inner automorphism group]] and is isomorphic to the [[quotient group]] <math>G/Z(G)</math> where <math>Z(G)</math> 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 <math>n</math>, the automorphism group is of order <math>\varphi(n)</math> where <math>\varphi</math> denotes the [[Euler totient function]]. Further, the automorphism group is cyclic iff <math>n</math> is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime <math>p</math>, the automorphism group of the cyclic group of order <math>p</math> is the cyclic group of order <math>p - 1</math>.|-| [[cyclic group:Z3]] || 3 || [[cyclic group:Z2]] || 2 || [[endomorphism structure of cyclic group:Z3]] || For a [[finite cyclic group]] of order <math>n</math>, the automorphism group is of order <math>\varphi(n)</math> where <math>\varphi</math> denotes the [[Euler totient function]]. Further, the automorphism group is cyclic iff <math>n</math> is 2,4, a power of an odd prime, or twice a power of ~~being ~~an odd prime. In particular, for a prime <math>p</math>, the automorphism group of the cyclic group of order <math>p</math> is the cyclic group of order <math>p - 1</math>.|-| [[cyclic group:Z4]] || 4 || [[cyclic group:Z2]] || 2 || [[~~inner ~~endomorphism structure of cyclic group:Z4]] || For a [[finite cyclic group]] of order <math>n</math>, the automorphism group is of order <math>\varphi(n)</math> where <math>\varphi</math> denotes the [[Euler totient function]]. Further, the automorphism group is cyclic iff <math>n</math> 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 <math>p^n</math>, the automorphism group is the [[general linear group]] <math>GL(n,p)</math>.|-| [[cyclic group:Z5]] || 5 || [[cyclic group:Z4]] || 4 || [[endomorphism structure of cyclic group:Z5]] || For a [[finite cyclic group]] of order <math>n</math>, the automorphism group is of order <math>\varphi(n)</math> where <math>\varphi</math> denotes the [[Euler totient function]]. Further, the automorphism group is cyclic iff <math>n</math> is 2,4, a power of an odd prime, or twice a power of an odd prime. In particular, for a prime <math>p</math>, the automorphism group of the cyclic group of order <math>p</math> is the cyclic group of order <math>p - 1</math>.|-| [[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 <math>n</math>, the automorphism group is of order <math>\varphi(n)</math> where <math>\varphi</math> denotes the [[Euler totient function]]. Further, the automorphism group is cyclic iff <math>n</math> 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 <math>p</math>, the automorphism group of the cyclic group of order <math>p</math> is the cyclic group of order <math>p - 1</math>.|-| [[cyclic group:Z8]] || 8 || [[Klein four-group]] || 4 || [[endomorphism structure of cyclic group:Z8]] || For a [[finite cyclic group]] of order <math>n</math>, the automorphism group is of order <math>\varphi(n)</math> where <math>\varphi</math> denotes the [[Euler totient function]]. Further, the automorphism group is cyclic iff <math>n</math> 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 <math>p^n</math>, the automorphismgroup is the [[general linear group]]<math>GL(n, p)</math>. In this case, <math>n = 3, p = 2</math>, so we get <math>GL(3,2)</math>, which [[isomorphism between linear groups over field:F2|is isomorphic to]] <math>PSL(3,2)</math>.|-| [[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 <math>S_n</math> is a [[complete group]] if <math>n \ne 2,6</math>.|} ===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 <math>n</math>, it is an abelian group of order <math>\varphi(n)</math> defined as the [[multiplicative group modulo n]]. It is itself cyclic if <math>n = 2,4</math>, 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]]|}

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