Outer linear group

Definition

In terms of the transpose-inverse map

The outer linear group of degree $n$ over a field $k$ is defined as the external semidirect product of the general linear group $GL(n,k)$ with a cyclic group of order two, where the non-identity element of the cyclic group acts by the transpose-inverse map

The definition also makes sense if the field $k$ is replaced by a commutative unital ring $R$.

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)	$336=2^{4}\cdot 3\cdot 7$