Outer linear group

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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