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On the number of subgroups and distinct fuzzy subgroups of a group defined by a presentation

Olusola B. Ogunfolu
기초자연과학연구소 (The Natural Science Institute Wonkwang University)|2023.10|ANNALS OF FUZZY MATHEMATICS AND INFORMATICS|제26권 제2호|pp. 189-198|DOI 10.30948/afmi.2023.26.2.189

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키워드

Fuzzy subgroupsSubgroup latticeFuzzy subgroup latticeChain of subgroups

초록

In this paper, we determined the number of subgroups of a group $ \rho_{2} \in G $ defined by a presentation $ \rho_{2} =\left\langle a,b \mid a^{27} = b^{27} = e, \left[ a,b \right] = a^{9} \right\rangle $, where $ a, b $ are generators and $ G $ is a class of finite minimal nonabelian p-groups with generators of the same order. The form and the order of elements of $ \rho_{2} $ were identified. We determined chains of subgroups and the number of distinct fuzzy subgroups by defining a natural equivalence relation on the subgroup lattice $ L(\rho_{2} \in G) $ associated with counting number of chains of subgroups of $ \rho_{2} $ that ends in $ \rho_{2}. $

Abstract

In this paper, we determined the number of subgroups of a group $ \rho_{2} \in G $ defined by a presentation $ \rho_{2} =\left\langle a,b \mid a^{27} = b^{27} = e, \left[ a,b \right] = a^{9} \right\rangle $, where $ a, b $ are generators and $ G $ is a class of finite minimal nonabelian p-groups with generators of the same order. The form and the order of elements of $ \rho_{2} $ were identified. We determined chains of subgroups and the number of distinct fuzzy subgroups by defining a natural equivalence relation on the subgroup lattice $ L(\rho_{2} \in G) $ associated with counting number of chains of subgroups of $ \rho_{2} $ that ends in $ \rho_{2}. $

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