On subpolygroup commutativity degree of finite polygroups
Probabilistic group theory is concerned with the probability of group elements or group subgroups satisfying certain conditions. On the other hand, a polygroup is a generalization of a group and a special case of a hypergroup. This paper generalizes probabilistic group theory to probabilistic polygr...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-08-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231211?viewType=HTML |
| _version_ | 1797742965443002368 |
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| author | M. Al Tahan Sarka Hoskova-Mayerova B. Davvaz A. Sonea |
| author_facet | M. Al Tahan Sarka Hoskova-Mayerova B. Davvaz A. Sonea |
| author_sort | M. Al Tahan |
| collection | DOAJ |
| description | Probabilistic group theory is concerned with the probability of group elements or group subgroups satisfying certain conditions. On the other hand, a polygroup is a generalization of a group and a special case of a hypergroup. This paper generalizes probabilistic group theory to probabilistic polygroup theory. In this regard, we extend the concept of the subgroup commutativity degree of a finite group to the subpolygroup commutativity degree of a finite polygroup $ P $. The latter measures the probability of two random subpolygroups $ H, K $ of $ P $ commuting (i.e., $ HK = KH $). First, using the subgroup commutativity table and the subpolygroup commutativity table, we present some results related to the new defined concept for groups and for polygroups. We then consider the special case of a polygroup associated to a group. We study the subpolygroup lattice and relate this to the subgroup lattice of the base group; this includes deriving an explicit formula for the subpolygroup commutativity degree in terms of the subgroup commutativity degree. Finally, we illustrate our results via non-trivial examples by applying the formulas that we prove to the associated polygroups of some well-known groups such as the dihedral group and the symmetric group. |
| first_indexed | 2024-03-12T14:47:47Z |
| format | Article |
| id | doaj.art-59aea7351f054def9314f6f12228453b |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-12T14:47:47Z |
| publishDate | 2023-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-59aea7351f054def9314f6f12228453b2023-08-16T01:39:33ZengAIMS PressAIMS Mathematics2473-69882023-08-01810237862379910.3934/math.20231211On subpolygroup commutativity degree of finite polygroupsM. Al Tahan 0 Sarka Hoskova-Mayerova1B. Davvaz2A. Sonea31. Department of Mathematics and Statistics, College of Arts and Science, Abu Dhabi University, United Arab Emirates2. Department of Mathematics and Physics, University of Defence, Brno, Czech Republic3. Department of Mathematical Sciences, Yazd University, Yazd, Iran4. Department of Sciences, University of Life Sciences, Iasi, RomaniaProbabilistic group theory is concerned with the probability of group elements or group subgroups satisfying certain conditions. On the other hand, a polygroup is a generalization of a group and a special case of a hypergroup. This paper generalizes probabilistic group theory to probabilistic polygroup theory. In this regard, we extend the concept of the subgroup commutativity degree of a finite group to the subpolygroup commutativity degree of a finite polygroup $ P $. The latter measures the probability of two random subpolygroups $ H, K $ of $ P $ commuting (i.e., $ HK = KH $). First, using the subgroup commutativity table and the subpolygroup commutativity table, we present some results related to the new defined concept for groups and for polygroups. We then consider the special case of a polygroup associated to a group. We study the subpolygroup lattice and relate this to the subgroup lattice of the base group; this includes deriving an explicit formula for the subpolygroup commutativity degree in terms of the subgroup commutativity degree. Finally, we illustrate our results via non-trivial examples by applying the formulas that we prove to the associated polygroups of some well-known groups such as the dihedral group and the symmetric group.https://www.aimspress.com/article/doi/10.3934/math.20231211?viewType=HTMLpolygroupsubgroup commutativity degreesubpolygroup latticesubpolygroup commutativity degree |
| spellingShingle | M. Al Tahan Sarka Hoskova-Mayerova B. Davvaz A. Sonea On subpolygroup commutativity degree of finite polygroups AIMS Mathematics polygroup subgroup commutativity degree subpolygroup lattice subpolygroup commutativity degree |
| title | On subpolygroup commutativity degree of finite polygroups |
| title_full | On subpolygroup commutativity degree of finite polygroups |
| title_fullStr | On subpolygroup commutativity degree of finite polygroups |
| title_full_unstemmed | On subpolygroup commutativity degree of finite polygroups |
| title_short | On subpolygroup commutativity degree of finite polygroups |
| title_sort | on subpolygroup commutativity degree of finite polygroups |
| topic | polygroup subgroup commutativity degree subpolygroup lattice subpolygroup commutativity degree |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231211?viewType=HTML |
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