Subgroup congruences for groups of prime power order

DOI10.1016/J.JALGEBRA.2022.08.002arXiv2201.07504MaRDI QIDQ6388627

Maria Loukaki, Stefanos Aivazidis

Publication date: 19 January 2022

Abstract: Given a $p$-group $G$ and a subgroup-closed class $mathfrakX$, we associate with each $mathfrakX$-subgroup $H$ certain quantities which count $mathfrakX$-subgroups containing $H$ subject to further properties. We show in Theorem I that each one of the said quantities is always $equiv1pmodp$ if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal $mathfrakX$-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.

Mathematics Subject Classification ID

Arithmetic and combinatorial problems involving abstract finite groups (20D60) Finite nilpotent groups, (p)-groups (20D15)

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