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

Sunday Adesina Adebisi; Mike Ogiugo; Michael Enioluwafe

doi:doi:10.11648/j.mcs.20210603.11

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

Sunday Adesina Adebisi, Mike Ogiugo, Michael Enioluwafe

Published in Mathematics and Computer Science (Volume 6, Issue 3)

Received: 8 May 2021 Accepted: 29 May 2021 Published: 7 June 2021

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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.

Published in	Mathematics and Computer Science (Volume 6, Issue 3)
DOI	10.11648/j.mcs.20210603.11
Page(s)	45-48
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Finite p-Groups, Abelian Group, Fuzzy Subsets, Fuzzy Subgroups, Inclusion-Exclusion Principle, Maximal Subgroups, Nilpotent Group

References

[1]	Adebisi S. A., M. Ogiugo and M. EniOluwafe (2020) Computing the Number of Distinct Fuzzy Subgroups for the Nilpotent p-Group of International J. Math. Combin. Vol. 1 (2020), 86-89.
[2]	Adebisi S. A., M. Ogiugo and M. EniOluwafe (2020) Determining The Number of Distinct Fuzzysubgroups For The Abelian Structure: Transactions of the Nigerian Association of Mathematical Physics Volume 11, (January - June, 2020 Issue), pp 5-6.
[3]	Adebisi S. A., M. Ogiugo and M. EniOluwafe (2020) The Fuzzy Subgroups For The Abelian Structure: Z8×Z2n; n >2. Journal of the Nigerian Mathematical Society, Vol. 39, Issue 2, pp. 167-171.
[4]	Adebisi S. A., M. Ogiugo and M. EniOluwafe Fuzzy Subgroups For (The Cartesian Product of) The Abelian Structure Journal of The Nigerian Mathematical Society (Submitted for publication).
[5]	Adebisi S. A., M. Ogiugo and M. EniOluwafe Fuzzy Subgroups For the Abelian Structurigerian Journal of Mathematics and Applications (Submitted).
[6]	Mashinchi M., Mukaidono M. (1992). A classification of fuzzy subgroups. Ninth Fuzzy System Symposium, Sapporo, Japan, 649-652.
[7]	Tarnauceanu Marius. (2013). Classifying fuzzy subgroups for a class of finite p-groups. “ALL CUZa” Univ. Iasi, Romania pp 30-39.
[8]	Tarnauceanu, Marius. (2012). Classifying fuzzy subgroups of finite nonabelian groups. Iran. J. Fussy Systems. (9) 33-43.
[9]	Tarnauceanu, M., Bentea, L.(2008). A note on the number of fuzzy subgroups of finite groups, Sci. An. Univ.”ALL. Cuza” Iasi, Matt., (54) 209-220.
[10]	Tarnauceanu, Marius., Bentea, L. (2008). On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems (159), 10841096, doi: 10.1016/j.fss.2017.11.014.
[11]	S. A. ADEBISI, M. Ogiugo and M. EniOluwafe (2020) NEW DISCOVERIES ON THE FINITE p-GROUPS OF ORDER: Transnational Journal of Mathematical Analysis and Applications Vol. 8, Issue 1, 2020, Pages 17-23 ISSN 2347-9086 Published Online on December 21, Jyoti Academic Press http://jyotiacademicpress.org.
[12]	S. A. ADEBISI, M. Ogiugo and M. EniOluwafe (2020) DETERMINING THE NUMBER OF DISTINCT FUZZYSUBGROUPS FOR THE ABELIAN STRUCTURE: Transactions of the Nigerian Association of Mathematical Physics Volume 11, (January - June, 2020 Issue), pp 5-6.
[13]	Akgual, M.(1988). Some Properties of Fuzzy Subgroups. J. Maths. Anal. Appl. 133, 93-100.
[14]	Blackburn, N. (1960). Nilpotent groups in which the derived group has two generators. J. Lond. Math. Soc., 35, 33-35.
[15]	Gagola, S. M. and Lewis, M. L. (1999). A Character theoretic condition characterizing nilpotent groups, Comm. Algebra 27 (3), 1053-1056.
[16]	Blackburn, N. (1965). Conjugacy in nilpotent groups. Proc. Amer. Math. Soc. 16, 143-148.

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APA Style

Sunday Adesina Adebisi, Mike Ogiugo, Michael Enioluwafe. (2021). The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups. Mathematics and Computer Science, 6(3), 45-48. https://doi.org/10.11648/j.mcs.20210603.11

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ACS Style

Sunday Adesina Adebisi; Mike Ogiugo; Michael Enioluwafe. The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups. Math. Comput. Sci. 2021, 6(3), 45-48. doi: 10.11648/j.mcs.20210603.11

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AMA Style

Sunday Adesina Adebisi, Mike Ogiugo, Michael Enioluwafe. The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups. Math Comput Sci. 2021;6(3):45-48. doi: 10.11648/j.mcs.20210603.11

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@article{10.11648/j.mcs.20210603.11,
  author = {Sunday Adesina Adebisi and Mike Ogiugo and Michael Enioluwafe},
  title = {The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups},
  journal = {Mathematics and Computer Science},
  volume = {6},
  number = {3},
  pages = {45-48},
  doi = {10.11648/j.mcs.20210603.11},
  url = {https://doi.org/10.11648/j.mcs.20210603.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20210603.11},
  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.},
 year = {2021}
}

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TY - JOUR
T1 - The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups
AU - Sunday Adesina Adebisi
AU - Mike Ogiugo
AU - Michael Enioluwafe
Y1 - 2021/06/07
PY - 2021
N1 - https://doi.org/10.11648/j.mcs.20210603.11
DO - 10.11648/j.mcs.20210603.11
T2 - Mathematics and Computer Science
JF - Mathematics and Computer Science
JO - Mathematics and Computer Science
SP - 45
EP - 48
PB - Science Publishing Group
SN - 2575-6028
UR - https://doi.org/10.11648/j.mcs.20210603.11
AB - 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.
VL - 6
IS - 3
ER -

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Author Information

Sunday Adesina Adebisi

Department of Mathematics, Faculty of Science, University of Lagos, Lagos, Nigeria
Mike Ogiugo

Department of Mathematics, School of Science, Yaba College of Technology, Lagos, Nigeria
Michael Enioluwafe

Department of Mathematics, Faculty of Science, University of Ibadan, Ibadan, Nigeria

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Plain Text BibTeX RIS

APA Style

Sunday Adesina Adebisi, Mike Ogiugo, Michael Enioluwafe. (2021). The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups. Mathematics and Computer Science, 6(3), 45-48. https://doi.org/10.11648/j.mcs.20210603.11

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ACS Style

Sunday Adesina Adebisi; Mike Ogiugo; Michael Enioluwafe. The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups. Math. Comput. Sci. 2021, 6(3), 45-48. doi: 10.11648/j.mcs.20210603.11

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AMA Style

Sunday Adesina Adebisi, Mike Ogiugo, Michael Enioluwafe. The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups. Math Comput Sci. 2021;6(3):45-48. doi: 10.11648/j.mcs.20210603.11

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@article{10.11648/j.mcs.20210603.11,
  author = {Sunday Adesina Adebisi and Mike Ogiugo and Michael Enioluwafe},
  title = {The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups},
  journal = {Mathematics and Computer Science},
  volume = {6},
  number = {3},
  pages = {45-48},
  doi = {10.11648/j.mcs.20210603.11},
  url = {https://doi.org/10.11648/j.mcs.20210603.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20210603.11},
  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.},
 year = {2021}
}

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