The chain structure of intuitionistic level subgroups in cyclic groups of order \(pq\)

Abstract

It is well known that the set of all level subgroups of any fuzzy subgroup of a finite group forms a chain. In this paper we prove that this result does not extend to Intuitionistic Fuzzy Subgroups (IFSGs) by providing a counter-example. For any two distinct prime numbers \(p\) and \(q\), we prove that the cyclic group \(\mathbb{Z}_{p q}\) has 36 non-isomorphic IFSGs. The Intuitionistic Level Subgroups (ILSGs) of only 28 of them form chains, while those of remaining 8 do not form chains. The list of all the 36 distinct IFSGs is also provided; and those whose ILSGs form a chain, and not, are identified. The case is illustrated using a specific example. We have also obtained a characterisation of IFSGs of \(\mathbb{Z}_{p q}\), whose ILSGs form a chain.