# Contents

## Definition

For $G$ a group, the outer automorphism group $\mathrm{Out}\left(G\right)$ is the quotient of the automorphism group by the normal subgroup of inner autormorphisms:

$\mathrm{Out}\left(G\right):=\mathrm{Aut}\left(G\right)/\mathrm{Inn}\left(G\right)\phantom{\rule{thinmathspace}{0ex}}.$Out(G) := Aut(G)/Inn(G) \,.

## Properties

### As a truncation of the automorphism 2-group

Write

$\mathrm{AUT}\left(G\right):={\mathrm{Aut}}_{\mathrm{Grpd}}\left(BG\right)$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

$G\stackrel{\mathrm{Ad}}{\to }\mathrm{Aut}\left(G\right)\phantom{\rule{thinmathspace}{0ex}}.$G \stackrel{Ad}{\to} Aut(G) \,.

Therefore the 0-truncation of $\mathrm{AUT}\left(G\right)$ is $\mathrm{Out}\left(G\right)$:

$\mathrm{Out}\left(G\right)\simeq {\tau }_{0}\mathrm{AUT}\left(G\right)\phantom{\rule{thinmathspace}{0ex}}.$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)