| For a collection of subgroups P of a finite group G, we define the counting function psiP(g) = |{ x ∈ G: ⟨g,x⟩ ∈ P }|. We partition the set of subgroups of G by defining an equivalence relation so that psiP is a generalized character for every equivalence class P. In some groups the equivalence relation can be refined to produce more generalized characters. We classify all refinements for elementary abelian p-groups of order p3, dihedral groups, and the direct product of a dihedral group of order 2 p and a cyclic group. |
