The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups
Date
2021
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Publisher
Science Publishing Group
Abstract
In the recent past, results have shown that Nilpotent groups such as p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian product (otherwise known as the product set) plays vital roles in the course of synthesizing the abstract groups. Previous
studies have determined the number of distinct fuzzy subgroups of various finite p-groups including those of square-free order.
However, not much work has been done on the fuzzy subgroup classification for the nilpotent groups formed from the
Cartesian products of p-groups through their computations. Here, part of our intention is therefore trying to make some designs
so as to classify the nilpotent groups formed from the Cartesian products of p-groups through their computations. The
Cartesian products of p-groups were taken to obtain nilpotent groups. Results up to two dimensions are now obtainable. In this
paper, the fuzzy subgroups of the nilpotent product of two abelian subgroups of orders 2n and 128. The integers n ≥ 7 have
been successfully considered and the derivation for the explicit formulae for its number distinct fuzzy subgroups were calculated. Some methods were once being used in counting the chains of fuzzy subgroups of an arbitrary finite p-group G. Here, the adoption of the famous Inclusion-Exclusion principle is very necessary and imperative so as to obtain a reasonable, and as much as possible accurate
Description
Keywords
Finite p-Groups, Abelian Group, Fuzzy Subsets, Fuzzy Subgroups, Inclusion-Exclusion Principle, Maximal Subgroups, Nilpotent Group