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- In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. An automorphism of a group which is not inner is called an outer automorphism. The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, An, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group, Sn, conjugation by any odd permutation is an outer automorphism of An or more precisely "represents the class of the (non-trivial) outer automorphism of An", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element. (en)
- 군론에서 외부자기동형군(外部自己準同型群, 영어: outer automorphism group)은 내부자기동형사상이 아닌 자기동형사상들로 이루어진 군이다. 그 원소를 외부자기동형사상(外部自己準同型寫像, 영어: outer automorphism)이라고 한다. (ko)
- In matematica si dice automorfismo esterno un automorfismo che non è un automorfismo interno, ovvero tale che non esiste alcun elemento del gruppo che possa indurre per coniugio l'automorfismo. Gli automorfismi esterni si possono ottenere come quoziente del gruppo degli automorfismi rispetto al sottogruppo normale degli automorfismi interni. (it)
- In de abstracte algebra is een uitwendig automorfisme van een groep elk automorfisme dat geen inwendig automorfisme is. (nl)
- 抽象代數的群論中，群G的外自同構群Out(G)是自同構群Aut(G)對內自同構群Inn(G)的商群Aut(G)/Inn(G)。 G的一個自同構如不是內自同構，便稱為外自同構。外自同構群Out(G)的元素是G的內自同構子群Inn(G)在自同構群Aut(G)中的陪集，故其元素不是外自同構，同一元素可對應到某個外自同構和任何內自同構的複合，因此不能定義G的外自同構群於G上的作用。不過因為內自同構都將群G的元素映射到同共軛類的元素，所以可定義出外自同構群在G的共軛類上的作用。 然而，若G為阿貝爾群，則G的內自同構群是平凡群，於是Out(G)可以自然地等同於Aut(G)，即是Out(G)的每個元素都對應唯一的自同構，因此Out(G)可以作用於G上。（而這時G的共軛類也各僅有一個元素。） (zh)
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- 군론에서 외부자기동형군(外部自己準同型群, 영어: outer automorphism group)은 내부자기동형사상이 아닌 자기동형사상들로 이루어진 군이다. 그 원소를 외부자기동형사상(外部自己準同型寫像, 영어: outer automorphism)이라고 한다. (ko)
- In matematica si dice automorfismo esterno un automorfismo che non è un automorfismo interno, ovvero tale che non esiste alcun elemento del gruppo che possa indurre per coniugio l'automorfismo. Gli automorfismi esterni si possono ottenere come quoziente del gruppo degli automorfismi rispetto al sottogruppo normale degli automorfismi interni. (it)
- In de abstracte algebra is een uitwendig automorfisme van een groep elk automorfisme dat geen inwendig automorfisme is. (nl)
- 抽象代數的群論中，群G的外自同構群Out(G)是自同構群Aut(G)對內自同構群Inn(G)的商群Aut(G)/Inn(G)。 G的一個自同構如不是內自同構，便稱為外自同構。外自同構群Out(G)的元素是G的內自同構子群Inn(G)在自同構群Aut(G)中的陪集，故其元素不是外自同構，同一元素可對應到某個外自同構和任何內自同構的複合，因此不能定義G的外自同構群於G上的作用。不過因為內自同構都將群G的元素映射到同共軛類的元素，所以可定義出外自同構群在G的共軛類上的作用。 然而，若G為阿貝爾群，則G的內自同構群是平凡群，於是Out(G)可以自然地等同於Aut(G)，即是Out(G)的每個元素都對應唯一的自同構，因此Out(G)可以作用於G上。（而這時G的共軛類也各僅有一個元素。） (zh)
- In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. (en)
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