# Outer linear group

## Definition

### In terms of the transpose-inverse map

The outer linear group of degree $n$ over a commutative unital ring $R$ is defined as the external semidirect product of the general linear group $GL(n,R)$ 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
$q$ 1 Dihedral group $D_{2(q-1)}$ $2(q-1)$ Multiplicative group of field is cyclic of order $q - 1$, outer automorphism acts by inverse map.
$2^n$ 2 Direct product of $SL(2,2^n)$ and dihedral group $D_{2(2^n - 1)}$
2 1 Cyclic group:Z2 $2$
2 2 Dihedral group:D12 (also, direct product of $S_3$ and $\mathbb{Z}_2$ $12 = 2^2 \cdot 3$ supersolvable but not nilpotent.
3 2 Outer linear group:OL(2,3) $96 = 2^5 \cdot 3$ solvable
4 2 Direct product of A5 and S3 $360 = 2^3 \cdot 3^2 \cdot 5$
5 2 Outer linear group:OL(2,5) $960 = 2^6 \cdot 3 \cdot 5$
2 3 Projective general linear group:PGL(2,7) $336 = 2^4 \cdot 3 \cdot 7$