The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups

Sunday Adesina Adebisi,Michael Enioluwafe,Mike Ogiugo

doi:10.11648/j.mcs.20210603.11

Sunday Adesina Adebisi, Michael Enioluwafe + Show 1 more

Open Access

https://doi.org/10.11648/j.mcs.20210603.11

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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.

Highlights

IntroductionTechniques and approaches have been used for the classification of which some are obtainable (see [6] and [10])
Many methods, techniques and approaches have been used for the classification of which some are obtainable
One particular case or the other have been treated by several papers such as the finite abelian as well as the non-abelian groups

Summary

Introduction

Techniques and approaches have been used for the classification of which some are obtainable (see [6] and [10]). Some elements of fuzzy logics come to play in the process of computing certain numbers of subgroups. In the recent few years, the problem of classifying the fuzzy subgroups of a finite group has experienced a very rapid and dynamic developments. In (Akgual, 1988) (see [13]), the number of fuzzy subgroups of a finite cyclic group of square-free order which are distinct has been. In (Gegang, 2005), a recurrence relation is indicated which can successfully be used to count the number of distinct fuzzy subgroups for two classes of finite abelian groups, namely: the arbitrary finite cyclic groups and finite elementary abelian p- groups. In 1771 had a theorem accredited to him based on finite Group He did not prove this theorems all he did, essentially, was to discuss some special cases. Berkovich and Zvonimir Janko who are from Israel and Germany respectively

The Historical Backgrounds of Logical Fuzziness

Modern Theory and Techniques of Logical Arithmetic

Fuzzy Sets

Basic Definitions and Terms

The Techniques

Conclusion