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:
for the automorphism 2-group of GG. This is the strict 2-group coming from the crossed module
Therefore the 0-truncation of AUT(G)AUT(G) is Out(G)Out(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.
automorphism group, automorphism ∞-group,
center, center of an ∞-group
outer automorphism group, outer automorphism ∞-group