No abstract

This article computes the number of fuzzy subgroups of symmetric group S 5 . First, an equivalence relation on the set of all fuzzy subgroups of a group G is defined.Without any equivalence relation on fuzzy subgroups of group G, the number of fuzzy subgroups is infinite, even for the trivial group. The Inclusion-Exclusion principle is used to determine the number of distinct fuzzy subgroups of symmetric group S 5 . Some inequalities satisfied by this number are also established for n ≥ 5

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 2 n 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.

A vanishing result is obtained in respect of nil groups of rings of finite global dimension. Also a connection is established with the extension problem.Mathematics Subject Classification: 55N15, 55U05

It is proved that the homology of an induced abelian group with coefficients in the different Gmodules occuring in its projective resolution are isomorphic.