# Outer linear group

From Groupprops

## Contents

## Definition

### In terms of the transpose-inverse map

The **outer linear group** of degree over a commutative unital ring is defined as the external semidirect product of the general linear group with a cyclic group of order two, where the non-identity element of the cyclic group acts by the transpose-inverse map.

## Particular cases

### Finite fields

Size of field | Degree (order of matrices) | Common name for the outer linear group | Order of group | Comment |
---|---|---|---|---|

1 | Dihedral group | Multiplicative group of field is cyclic of order , outer automorphism acts by inverse map. | ||

2 | Direct product of and dihedral group | |||

2 | 1 | Cyclic group:Z2 | ||

2 | 2 | Dihedral group:D12 (also, direct product of and | supersolvable but not nilpotent. | |

3 | 2 | Outer linear group:OL(2,3) | solvable | |

4 | 2 | Direct product of A5 and S3 | ||

5 | 2 | Outer linear group:OL(2,5) | ||

2 | 3 | Projective general linear group:PGL(2,7) |