Subgroup congruences for groups of prime power order

TL;DR

This paper investigates subgroup congruences in p-groups, establishing conditions under which certain subgroup counting quantities are congruent to 1 modulo p, and extends classical results like Burnside's theorem for abelian groups.

Contribution

It introduces new subgroup counting invariants for p-groups and proves their congruence properties, extending classical theorems and providing new corollaries.

Findings

Quantities counting certain subgroups are always ≡ 1 mod p under specific conditions.

Normal subgroup cases are characterized with additional properties.

A sharpened version of Burnside's theorem for abelian groups of bounded exponent is obtained.

Abstract

Given a $p$-group $G$ and a subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing $H$ subject to further properties. We show in Theorem I that each one of the said quantities is always $\equiv 1 \pmod p$ if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal $\mathfrak{X}$-subgroups and in Theorem III we obtain a sharpened version of a celebrated theorem of Burnside relative to the class of abelian groups of bounded exponent. Various other corollaries are also presented.