The real genus of \(p\)-groups (Q2861062)

The \textit{real genus} \(\rho(G)\) of a finite group \(G\) is the smallest algebraic genus of the compact bordered Klein surfaces on which \(G\) acts faithfully as a group of automorphisms. In a previous paper [Rocky Mt. J. Math. 37, No. 4, 1251--1269 (2007; Zbl 1134.30034)], the first author proved that if \(G\) is a \(p\)-group where \(p\) is an odd prime, then either \(\rho(G) < 2\) or \(\rho(G) \equiv p + 1\) mod \(2p\). In this paper the authors give a refinement of the latter theorem, by showing that if \(G\) is a non-cyclic \(p\)-group then either \(\rho(G) \equiv p^{3} + 1\) mod \(2p^3\) or \(G\) lies in one of a small number of exceptional families. They also consider the density of the subset of all positive integers that can occur as \(\rho(G)\) for some \(G\), and state some open problems at the end.