Abstract
In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.
Highlights
All of the groups considered in this paper are finite
Some classes of special subgroup chains are studied in several papers, like, in [4], the authors count the number of maximal subgroup chains of nilpotent groups
In [7,8], the authors study the well-known Delannoy numbers, and prove that they are just the numbers of the subgroup chains of a cyclic group that satisfy a certain property. Another interesting question is the classifying of distinct fuzzy subgroups of abelian groups, which can be translated into a combinatorial problem on the subgroup lattice of a group G: counting the number of some kind of subgroup chains of G
Summary
All of the groups considered in this paper are finite. Basic notations and concepts correspond to [1,2,3]. Some classes of special subgroup chains are studied in several papers, like, in [4], the authors count the number of maximal subgroup chains of nilpotent groups. This topic has close connection with some open questions of other fields of mathematics. In [7,8], the authors study the well-known Delannoy numbers, and prove that they are just the numbers of the subgroup chains of a cyclic group that satisfy a certain property Another interesting question is the classifying of distinct fuzzy subgroups of abelian groups, which can be translated into a combinatorial problem on the subgroup lattice of a group G: counting the number of some kind of subgroup chains of G. We make two specific examples of our applications at the end of the paper
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