The genus of the product of a group with an Abelian group (Q1824630)

The genus \(\gamma\) (G) of a group G is the minimum genus of any orientable surface containing an embedding of some Cayley graph for G. The genus of the direct product \(G\times A\) of an arbitrary finite group G and a finite abelian group A is determined for many G and most abelian groups A of sufficiently large rank. The authors present sufficient conditions on the groups G and A such that \(\gamma (G\times A)=1+| G\times A| (r-2)/4,\) where r is the rank of A, \(r>3\), and A has no factors \({\mathbb{Z}}_ 2\) and \({\mathbb{Z}}_ 3\) in its canonical form.