Subgroup congruences for groups of prime power order

Given a p-group G and a subgroup-closed class X, we associate with each X-subgroup H certain quantities which count X- subgroups containing H subject to further properties. We show in Theorem I that each one of the said quantities is always equivalent to 1 (mod p) if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal 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 and some open questions are posed. (C) 2022 Elsevier Inc. All rights reserved.