ISSN 1937 - 1055 V O L U M E 1, 2020 INTERNATIONAL JOURNAL OF M ATHEMATICAL COMBINATORICS EDITED B Y THE MADIS OF CHINESE ACAD EM Y OF SCIENCES AND ACAD EM Y OF M ATHEM ATICAL COMBINATORICS & APPLICATIONS, USA March, 2020 IBADAN UNIVERSITY LIBRARY V o l.l , 2020 ISSN 1937-1055 International Journal of Mathematical Combinatorics (www.mathcombin.com) Edited By The Madis of Chinese Academy of Sciences and Academy of Mathematical Combinatorics & Applications, USA March, 2020 IBADAN UNIVERSITY LIBRARY A im s and S cope: The mathematical combinatorics is a subject that applying combinatorial notion to all mathematics and all sciences for understanding the reality of things in the universe, motivated by CC Conjecture o f Dr.Linfan MAO on mathematical sciences. 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Zhang Department of Computer Science Georgia State University, Atlanta, USA Famous Words: We know nothing of what will happen in future, but by the analogy of past experience. By Abraham Lincoln, an American president IBADAN UNIVERSITY LIBRARY International ./. Math. Combin. Vol. 1(2020), 86-89 Computing the Number of Distinct Fuzzy Subgroups for the Ninpotent p-Group of D2» x C\ S. A. Adebisi1, M. Ogiugo 2 and M. EniOluwafe3 1. Department of Mathematics, Faculty of Science, University of Lagos, Nigeria 2. Department of mathematics,School of Science, Yaba College of Technology, Lagos, Nigeria 3. Department of mathematics,Faculty of Science, university of Ibadan, Nigeria E-mail: adesinasunday@yahoo.com, ekpenogiugo@gmail.com, michacl.enioluwafe@grnail.com A bstract: In this paper, the explicit formulae is given for the number of distinct fuzzy subgroups of the cartesian product of the dihedral group of order 2r‘ with a cyclic group of order four, where n > 3. K ey W ords: Finite p-Groups, nilpotent Group, fuzzy subgroups, dihedral Group, inclusion-exclusion principle, maximal subgroups. A M S (2 0 1 0 ): 20D15, G0A86. §1. In trod u ction In the fuzzy group theory , the classification of the fuzzy subgroups, most especially the finite p-groups cannot be underestimated. This aspect of pure Mathematics has undergone a dynamic developments over the years. For instance, many researchers have treated cases of finite abelian groups (see [2], [3]). The starting point for this concept all started as presented in [5] and [6] . Since then, the study has been extended to some other important classes of finite abelian and nonabelian groups such as the dihedral , quaternion, semidihedral, and hamiltonian groups. Although, the natural equivalence relation was introduced in [7], where a method to deter­ mine the number and nature of fuzzy subgroups of a finite group G was developed with respect to the natural equivalence. In [1] and [3], a different approach was applied for the classifica­ tion. In this work , an essential role in solving counting problems is played by adopting the Inclusion-Exclusion Principle. The process leads to some recurrence relations from which the solutions are then finally computed with ease. §2. P relim inaries Suppose that (G ,-,e ) is a group with identity e. Let S(G) denote the collection of all fuzzy subsets of G, An element A G S(G) is said to be a fuzzy subgroup of G if 1Receivcd September 29, 2019, Accepted March 15, 2020. IBADAN UNIVERSITY LIBRARY Computing the Number o f Distinct Fuzzy Subgroups for the Ninpotcnt p-Group of D2n X C\ 8 7 (i) \(ab) > m in{\ (a ), \ (b )}, V a,b € G\ (ii) A(a“ 1) > A (a) for any a € G. And, since (a -1 ) -1 = a, we have that A(n_1) = A(a), for any a 6 G. Also, by this notation and definition, A(e) = sup A(G) (see Marius [6]), which implies T h eorem 2.1 The set F L (G ) possessing all fuzzy subgroups o f G forms a lattice under the usual ordering of fuzzy set inclusion. This is called the fuzzy subgroup lattice of G. We define the level subset AGp = {a £ G/\(a) > /?} for each /3 £ [0,1] The fuzzy subgroups of a finite p-group G are thus, characterized, based on these level subsets. In the sequel, A is a fuzzy subgroup of G if and only if its level subsets are subgroups in G. Theorem 2.1 gives a link between F L (G ) and L(G), the classical subgroup lattice of G. Moreover, some natural relations on S {G ) can also be used in the process of classifying the fuzzy subgroups of a finite p-group G (see [6]). One of them is defined by: A ~ 7 if and only if (A(a) > A(b) ■$=> v(a) > v[b), V a,b £ G). Also, two fuzzy subgroups A, 7 of G and said to be distinct if A x v. As a result of this development, let G be a finite p-group and suppose that A : G — > [0,1] is a fuzzy subgroup of G. Put A(G) = ■ ■ ■ ,Pk} w'th the assumption that < fa > • • • > f3k- Then, ends in G is determined by A. AGpi C AG02 C • • • C AGPk = G ( 2 - 1 ) Also, we have that A(a) = 0t «=> t = m ax{r/a e AGpr) <=> a e AGp,\\G0l_1, for any a € G and t = 1, • • • , k, where by convention, set AGp0 — 4>. §3. M eth od o log y In the sequel, the method that will be used in counting the chains of fuzzy subgroups of an arbitrary finite p-group G is described. Suppose that M\, M 2, ■. ■ ,M t are the maximal subgroups of G, and denote by h(G) the number of chains of subgroups of G which ends in G. By simply applying the technique of computing h(G), using the application of the inclusion- exclusion principle, .we have that: h(G) = 2 ( £ h(Mr) - K Mrx n M ra) + • • • + ( - l ) t -1 /i ( p| Mr\ j . ( 3 - 1 ) y r = l \ h(Z4 x Z4) = 48. □ C orollary 4.2 Following Lemma 4.1, h(Z4 x Z 2s), h (Z4 x Z 2c), h(Z 4 x Z 2t) and h (Z4 x Z 2s) = 1536,4096,10496 and 26112, respectively. P rop osition 4.3 Suppose that G = Z 4 x Z 2« , n > 2. Then, h(G) = 2r‘ [n2 + 5n — 2], Proof G has three maximal subgroups of which two are isomorphic to Z 2 x Z 2« and the third is isomorphic to Z 4 x Z 2™-i. Hence, h(Z4 x Z 2n) — 2/i (Z 2 x Z 2« ) 4- 2̂ h (Z2 x Z 2n - 1) -t 22/i(Z2 x Z 2«-2 ) + 2’̂ /i.(Z2 x Z 2™ - 3) + 24/j,(Z2 x Z2n-j) + • • • + 2n~2/i(Z2 x Z 22) n -2 = 2" + 1[2( n + l ) + ^ ( ( n + l ) - i ) i=i = 2,l+1(2(n + 1) + i ( n - 2)(n + 3)) = 2"(n 2 + 5n - 2) for n > 2. We therefore know that /i(Z4 x Z 2n-i) = 2n -1 ((n — l )2 + 5(n — 1) — 2) = 2n_1(n2 + 3n — 6) for n > 2. This completes the proof. □ T h eorem 4.4([4]) Let G = D 2n x C2, f/ie nilpotent group formed by the- cartesian product of the dihedral group of order 2" and a cyclic group of order 2. Then, the number o f distinct fuzzy subgroups of G is given by h(G) = 22n(2n + 1) — 2n+1 for n > 3. IBADAN UNIVERSITY LIBRARY Computing the Number o f Distinct Fuzzy Subgroups for the Ninpotent p-Group of D 2™ X C 4 8 9 §5. T h e N u m ber o f Fuzzy Subgroups for D y x C4 P rop osition 5.1 Suppose that G = D 2n x C4. Then, the number of distinct fuzzy subgroups of G is given by n —3 22(n~2)(64n + 173) + 3 2(" _ 1+ j)(2n + 1 - 2j ) i =1 fo r n > 3. Proof Calculation shows that -lh (D 2" x Ci) = h(D2« x C2) + 2h(D2n-i x C4) - 4h(D2„ - i x C2) + /i(Z.1 x Z2„_i) — 2 /l(^2 x Zgw —l ) — 2/l(Z4 x Z g n —2) + Sh(Z2 X Z<2n. — 2) “h h{Zj2‘ n — 1 ) — Ah(Zj2n— 2) = (n - 3)22n+2 + 22(rl_3)(14G0) + 3(2n(2n - 1) + 2n+1(2ri - 3) + 2n+2(2n - 5) + • • • + 7(22(n_2))) n —3 = (n - 3).22n+2 + 22(n_3)(1460) + 3 2n- ' +J(2n + 1 - 2j) j = 1 n —3 = 22(n' 2)(64n + 173) + 3 ^ 2"_1+;’ (2n + 1 - 2j). This completes the proof. □ R eferences [1] Mashinchi M., Mukaidono M. , A classification of fuzzy subgroups, Ninth Fuzzy System Symposium (1992), 649-652. [2] Mashinchi M., Mukaidono M., On fuzzy subgroups classification, Research Reports of Meiji Univ., 9(1993), 31-36. [3] Murali V., Makamba B. B., On an equivalence of fuzzy subgroups III, Int. J. Math. Sci., 36(2003), 2303- 2313. [4] Adebisi S.A., Enioluwafe M., An explicit formula for the number of distinct fuzzy subgroups of the cartesian product of the Dihedral group of order 2n with- a cyclic group of order 2, Universal J.of Mathematics and Mathematical Sciences, Vol.13, No. 1(2020), 1-7. [5] Tarnauceanu M., The classification of fuzzy subgroups of finite cyclic groups and Delannoy number, European J.Combin. 30(2009), 283-289. [6] Tarnauceanu M., Classifying fuzzy subgroups of finite nonabelian groups, ,7. Fuzzy Syst., 9(2012), 33 - 43. [7] Tarnauceanu M., Classifying fuzzy subgroups for a class of finite p-groups, “ALL CUZa” Univ. Iasi,(2011). [8] Tarnauceanu M., Bentea L., On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems, 159(2008), 1084-1096. IBADAN UNIVERSITY LIBRARY