nLab
outer automorphism

Contents

Definition

For GG a group, the outer automorphism group Out(G)Out(G) is the quotient of the automorphism group by the normal subgroup of inner autormorphisms:

Out(G):=Aut(G)/Inn(G). Out(G) := Aut(G)/Inn(G) \,.

Properties

As a truncation of the automorphism 2-group

Write

AUT(G):=Aut Grpd(BG) AUT(G) := Aut_{Grpd}(\mathbf{B}G)

for the automorphism 2-group of GG. This is the strict 2-group coming from the crossed module

GAdAut(G). G \stackrel{Ad}{\to} Aut(G) \,.

Therefore the 0-truncation of AUT(G)AUT(G) is Out(G)Out(G):

Out(G)τ 0AUT(G). Out(G) \simeq \tau_0 AUT(G) \,.

This perspective generalizes to the notion of outer automorphism ∞-group.

Examples

The connected components of the subgroup of outer automorphisms of the a super Poincaré group which fixes the underlying Poincaré group is known as the R-symmetry group in supersymmetry.

Revised on August 28, 2013 17:45:08 by Urs Schreiber (82.113.98.24)