The classification of some fuzzy subgroups of finite groups under a natural equivalence and its extension, with particular emphasis on the number of equivalence classes

摘要

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .