Advanced Studies: Euro-Tbilisi Mathematical Journal

On k-invariants of crossed modules corresponding to the dihedral groups

G. Donadze, T. Pirashvili

Any group G gives rise to a crossed module ∂ : G → Aut(G), where ∂(g) is the inner automorphism defined by x ↦ gxg^-1. This crossed module is denoted by AUT(G). The first k-invariant of the classifying space of AUT(G) provides the class k^AUT(G) ∈ H³(Out(G),Z(G)), where Z(G) is the centre of G and Out(G) is the group of outer automorphisms of G. The goal of this work is to investigate whether this class is trivial for dihedral groups and their generalizations. Specifically, we compute the class k^AUT(D_n) explicitly when n is power of two.


Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 18(3) (2025), pp. 41-59