The symmetric crosscap number of a group (Q2777595)

Every finite group \(G\) acts effectively on some non-orientable surface. The minimal (non-orientable) genus of such a surface is called the symmetric crosscap number of the group \(G\), as the genus of a non-orientable surface is defined to be the number of crosscaps needed to construct it (i.e. the number of projective planes in connected sum that give the surface). NEWLINENEWLINENEWLINETucker classified the groups with crosscap number 1 or 2 and conjectured that there are no groups with crosscap number 3. This conjecture is proved here. The author also obtains some bounds on this number, as well as explicit formulas for three families of groups.