# Difference between revisions of "Automorphism group of a group"

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==Definition== | ==Definition== | ||

## Revision as of 00:22, 31 December 2007

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: (factscloselyrelated 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

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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 .