# GAP:AutomorphismGroup

This article is about a GAP function.

This GAP function takes as input a group. See more functions like this.

## Definition

### Function type

This GAP function takes as input a group and outputs a group, *along with* a stored action of that group on the original group.

### Behavior

For an input group, the function outputs the automorphism group of that group.

## Related functions

- GAP:Endomorphisms: This function also requires the Sonata package and outputs all the endomorphisms as maps from the group to itself.

## Examples of usage

### Some examples involving pre-specified groups

gap> G := SymmetricGroup(6); Sym( [ 1 .. 6 ] ) gap> H := AutomorphismGroup(G); <group with 3 generators> gap> Order(H); 1440

This snippet computes the automorphism group of the symmetric group on six letters.

gap> L := List(AllSmallGroups(8),AutomorphismGroup); [ <group with 2 generators>, <group with 4 generators>, <group of size 8 with 3 generators>, <group of size 24 with 4 generators>, <group with 4 generators> ] gap> List(L,IdGroup); [ [ 4, 2 ], [ 8, 3 ], [ 8, 3 ], [ 24, 12 ], [ 168, 42 ] ]

This snippet uses the AllSmallGroups command and List command to compute the list of automorphism groups of all groups of order eight. Next, it lists the group IDs of all these automorphism groups using the IdGroup command. The first number in each pair represents the order of the automorphism group, while the second number represents its position among the list of all groups of that order. In particular, this list shows that the automorphism groups of the five groups of order eight have sizes respectively.