Automorphism group of a group: Difference between revisions
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The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>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>Sym(G)</math>, the group of all permutations on <math>G</math>. | The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>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>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 are the property of being an [[inner automorphism]], [[class automorphism]], [[extensible automorphism]]. | |||
Revision as of 00:31, 31 December 2007
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 , denoted , is a set whose elements are automorphisms , and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of , the group of all permutations on .
Subgroups
Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Examples are the property of being an inner automorphism, class automorphism, extensible automorphism.