Enumeration of lifts of commuting elements of a group (original) (raw)

This paper discusses the enumeration of lifts of commuting elements within the framework of group theory. Building upon the foundational work of Frobenius and Mednykh, it extends the Frobenius-Mednykh formula to more general contexts through the study of group epimorphisms and homomorphisms related to surfaces. It presents a detailed examination of extra-special p-groups and characterizes the number of commuting elements in the permutations of these groups. Through examples and theorems, it highlights the significance of polynomial representations in determining the structure of lifts, contributing to the larger body of knowledge on commutation relations and representation theory.