An arithmetic method of counting the subgroups of a finite abelian group

By Marius Tùarnùauceanu. 2018

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One of the most important problems of the combinatorial abelian group theory is to determine the number of subgroups of a finite abelian group. This topic has enjoyed a constant evolution starting with the first half of the 20 century. Since a finite abelian group is a direct product of abelian p-groups, the above counting problem is reduced to p-groups. Formulas which give the number of subgroups of type μ of a finite p-group of type λ were established by S. Delsarte (see [7]), P.E. Djubjuk (see [8]) and Y. Yeh (see [15]). An excellent survey on this subject together with connections to symmetric functions was written by M.L. Butler (see [5]) in 1994. Another way to find the total number of subgroups of finite abelian p-groups is presented in [6] and applied for rank two p-groups, as well as for elementary abelian p-groups. Also, remind here the paper [1] which gives an explicit formula for the number of subgroups in a finite abelian p-group by using divisor functions of matrices.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

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