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Definition

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

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

Properties

As a truncation of the automorphism 2-group

Write

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

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

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

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

$Out(G) \simeq \tau_0 AUT(G) \,.$

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 July 16, 2014 08:37:55 by Urs Schreiber (89.204.138.8)