nLab outer automorphism




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 autormorphism?s:

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


As a truncation of the automorphism 2-group


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.


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.

Last revised on July 16, 2014 at 08:37:55. See the history of this page for a list of all contributions to it.