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.
Categories: math.GR