nLab outer automorphism

Definition

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

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

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.

Related concepts

• group, ∞-group,

• automorphism group, automorphism ∞-group,

• center, center of an ∞-group

• outer automorphism group, outer automorphism ∞-group

• Outer space