scholarly works

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Subject: search.filters.subject.Inclusion-Exclusion Principle

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    The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups
    (Science Publishing Group, 2021) Adebisi, S.A.; Ogiugo, M.; EniOluwafe, M.
    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
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    On the Nilpotent Fuzzy Subgroups of the Abelian Type: Z32 × Z2n , n ≥ 5
    (2020) Adebisi, S.A.; Ogiugo, M.; EniOluwafe, M.
    The extension of the finite nilpotent groups is now being diversified. As such, results up to two dimensions are now obtainable. In this paper, the fuzzy subgroups of the nilpo tent abelian structure given by: Z32 × Z2n , the cartesian product of two abelian subgroups of orders 2n and 32 respec tively for every integer n > 5 have been carefully studied and the explicit formulae for its number distinctly given.
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    FUZZY SUBGROUPS FOR (THE CARTESIAN PRODUCT OF) THE ABELIAN STRUCTURE : Z16×Z2n, n > 3
    (2020) Adebisi, S.A.; Ogiugo, M.; EniOluwafe, M.
    As part of the extension of the finite nilpotent groups to the direct product of p-groups, we give in this paper, the explicit formulae for the number of distinct fuzzy subgroups of the abelian structure given by: ℤ16 × 2, n > 3, the Cartesian product of two abelian groups of orders 2n and 16 respectively for every integer n > 3
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    The fuzzy subgroups for the abelian structure Z8 x Z2n , n > 2
    (Nigerian Mathematical Society, 2020) Adebisi, S. A.; Ogiugo, M.; EniOluwafe, M.
    Any finite nilpotent group can be uniquely written as a direct product of p-groups. In this paper, we give explicit formulae for the number of distinct fuzzy subgroups of the cartesian product of two abelian groups of orders 2n and 8 respectively for every integer n > 2 .
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    An explicit formula for the number of distinct fuzzy subgroups of the cartesian product of the dihedral group of order 2n with a cyclic group of order 2
    (Pushpa Publishing House, Prayagraj, India, 2020) Adebisi, S. A.; EniOluwafe, M.
    The problem of classification of fuzzy subgroups can be extended from finite p-groups to finite nilpotent groups. Accordingly, any finite nilpotent group can be uniquely written as a direct product of p-groups. In this paper, we give explicit formulae for the number of distinct fuzzy subgroups of the Cartesian product of the dihedral group of order 2n with a cyclic group of order 2.
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    Determining the number of distinct fuzzy subgroups for the abelian structure Z4 x Z2n-1 ,n > 2
    (2020) Adebisi, S.A.; Ogiugo, M.; EniOluwafe, M.
    The problem of classification of fuzzy subgroups can be extended from finite p-groups to finite nilpotent groups. Accordingly, any finite nilpotent group can be uniquely written as a direct product of p-groups. In this paper, we give explicit formulae for the number of distinct fuzzy subgroups of the Cartesian product of two abelian groups of orders 2n−1 and 4 respectively for every integer n >2.