Stiff genus of groups (Q2713635)

Orientable stiff embeddings of an \(n\)-coloured digraph arise from stiff arc rotation systems, which are those arc rotation systems where the rotations are determined up to their orientation by a fixed cyclic permutation of the signed colour set (adjacent vertices yield inverse permutations). Nonorientable stiff embeddings are defined similarly using the technique of embedding schemes. Considering all Cayley digraphs and minimizing the genus, one defines the orientable and the nonorientable stiff genus of a finite group \(G\). A nonorientable stiff embedding for a non-trivial \(G\) always exists, but the orientable one exists if and only if \(G\) has a subgroup of index two. The stiff genera are computed for elementary abelian groups, dihedral groups, generalized quaternions and non-abelian groups of order \(p^3\).