Menon-type identities derived from actions of subgroups of general linear groups

In this article, we consider the actions of subgroups of the general linear group GLr(Zn) on Znr, including groups of upper triangular matrices in GLr(Zn), unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy-Frobenius-Burnside lemma, we obtain several generalizations of the well known Menon's identity.