International Journal of Algebra, Vol. 6, 2012, no. 25, 1221 - 1225
On Subgroups Lattice of Quasidihedral Group Bashir Humera and Zahid Raza Department of Mathematics National University of Computer & Emerging Sciences Lahore Campus, Pakistan
[email protected],
[email protected] Abstract In this paper, we have determined the number of subgroups of finite non-abelain group defined by a presentation G = x, y | x2 = y 16 = 1, yx = xy 7 . We identify the form and the order of elements of the group G. There are 26 non-trivial subgroups of G and the subgroups lattice diagram is then presented.
Mathematics Subject Classification: 20B30, 20B35, 03G10 Keywords: Lagrange theorem, lattice, non-abelian graoup
1
Introduction
One of the most important problem of fuzzy theory is to classify the fuzzy subgroups of a finite group[4]. Several problems have treated the particular case finite abelian group. Laszlo studied the construction of fuzzy subgroups of groups of the orders one to six. Zhang and Zhou[15] determined the number of fuzzy subgroups of cyclic groups of the order pn where p is a prime number. Murali and Makamba [8], considering a similar problem, found the number of fuzzy subgroups of groups of the order pn q m where p and q are distinct primes under different equivalence relation. In [14], Tarnauceanu and Bentea established the recurrence relation verified by the number of fuzzy subgroups of finite cyclic groups. Their result improves Murali’s work in [10, 9]. In [12] the authors calculated the number of fuzzy subgroups of finite abelain groups. Since all of them have worked on finite abelian groups.It is interesting to investigate the number of fuzzy subgroups of nonabelian groups. Raden Sulaiman and Abd Ghafur Ahmed [4], worked on nonabelain groups S2 , S3 and A4 and constructed their fuzzy subgroups.The number of fuzzy subgroups of S4 is also calculated [13, 7].We have determined the subgroup lattice diagram of group G defined by the presentationG = x, y | x2 = y 16 = 1, yx = xy 7 which will help in determining number of fuzzy subgroups of G.
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B. Humera and Z. Raza
Preliminaries
In this section, we review the definitions and preliminary results that are required later in this paper. Throughout the section, a group G is assumed to be a finite non-abelian group defined by presentation G = x, y | x2 = y 16 = 1, yx = xy 7 and it is also called Quasidihedral group. Definition 2.1 [11] A partial ordered on a non-empty set P is a binary relation ≤ on P that is reflexive,antisymmetric and transitive.The pair P, ≤ is called a partially ordered set or poset.Poset P, ≤ is totally ordered if every x, y ∈ P are comparable, that is x ≤ y or y ≤ x. A non-empty subset S of P is a chain in P if S is totally ordered by ≤. Definition 2.2 [11] Let P, ≤ be a poset and let S ⊆ P . An upper bound for S is an element x ∈ P for which s ≤ x, ∀s ∈ S.The least upper bound of S is called the supremum or join of S.A lower bound of S is called the supremum or join of S.a lower bound of S is an element x ∈ P for which x ≤ s, ∀s ∈ S. The greatest lower bound of S is called the infimum or meet of S. The greatest lower bound of S is called the infimum or meet of S. Poset P, ≤ is called a lattice if every pair x, y elements of P has a supremum and an infimum. Definition 2.3 [5](Lagrange Theorem)If G is a finite group and H is a subgroup of G, then order of H is a divisor of order G. Definition 2.4 [5] If G is a finite group and a ∈ G, then order of a is a divisor of order of G. Definition 2.5 [6](The First Sylow Theorem) Let G be a finite group and let |G| = pn m where n ≤ 1,p is a prime number and (p, m) = 1. Then G contains a subgroup of order p for each i where 1 ≤ i ≤ n
3
Main Results
Consider G = x, y | x2 = y 16 = 1, yx = xy 7 be a non-abelian group, then the elements of G can be written as x, y, y 2, y 3 , . . . , y 15, xy, xy 2, xy 3 , . . . , xy 15 . The order of elements of G are given below: order 1 2 4 8 16
Elements {e} x, y 8, xy 2 , xy 4 , xy 6, xy 8 , xy 10 , xy 12 , xy 14 y 4, y 12 , xy, xy 3, xy 5, xy 7 , xy 9 , xy 11 , xy 13 , xy 15 y 2, y 6, y 10 , y 14 y, y 3, y 5, y 7 , y 9, y 11, y 13 , y 15
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On subgroups lattice of non-abelian group
According to the Lagrange theorem, non-trivial subgroups of G must have order 2, 4, 8, 16. We will determine all subgroups of G. Clearly, the subgroup of order 1 is the trivial subgroup H1 = e. The order of non-trivial subgroups of G are given by : Subgroups of order 2: Let H be a subgroup of order 2. Since 2 is a prime number, then H is cyclic. Therefore, H is generated by an element of G of order 2. Thus, all subgroups of order 2 are generated by elements of order 2 i.e., x, y 8, xy 8, xy 2, xy 4, xy 6, xy 10 , xy 12, xy 14 . Subgroups of order 4: Let M be any arbitrary subgroup of G of order 4. According to the theorem 2.4, the elements of M must have order 1, 2 or 4. If M consists an element of order 4, then M is a cyclic group. The generators of these subgroups are M1 = y 4 = y 12 = {e, y 4, y 8, y 12 },M2 = xy 3 = xy 11 = {e, xy 3 , y 8, xy 11 },M3 = xy 7 = xy 15 = {e, xy 7 , y 8, xy 15 },M4 = xy = xy 9 = {e, xy, y 8, xy 9 },M5 = xy 5 = xy 13 = {e, xy 5 , y 8, xy 13 }. If M is generated by an elements of order 2, then M does not have elements of order 4. Therefore, the order of elements of M is 2, except identity element. We will try to multiply all of the combinations of elements of order 2. The following is the multiplication table of elements of order 2. e x y8 xy 2 xy 4 xy 6 xy 8 xy 10 xy 12 xy 14
x e xy 8 y 14 y 12 y 10 y8 y6 y4 y2
y8 xy 8 e xy 10 xy 12 xy 14 x xy 2 xy 4 xy 6
xy 2 y2 xy 10 e y 14 y 12 y 10 y8 y6 y4
xy 4 y4 xy 12 y2 e y 14 y 12 y 10 y8 y6
xy 6 y6 xy 14 y4 y2 e y 14 y 12 y 10 y8
xy 8 y8 x y6 y4 y2 e y 14 y 12 y 10
xy 10 y 10 xy 2 y8 y6 y4 y2 e y 14 y 12
xy 12 y 12 xy 4 y 10 y8 y6 y4 y2 e y 14
xy 14 y 14 xy 6 y 12 y 10 y8 y6 y4 y2 e
We observe that the subgroups of order 4 generated by elements of order 2 are M6 = x, y 8 = x, xy 8 = {e, x, y 8 , xy 8}, M7 = xy 2 , y 8 = xy 10 , xy 8 = {e, xy 2 , y 8, xy 10 }, M8 = xy 4, y 8 = xy 12 , y 8 = {e, xy 4 , y 8, xy 12 }, M9 = xy 6 , y 8 = xy 14 , y 8 = {e, xy 6 , y 8, xy 14 }. Thus we have 9 subgroups of order 4. Subgroups of order 8: Let N be any arbitrary subgroup of G of order 8. If N contains an element of order 8, then N is cyclic. The elements of order 8 i.e.,y 2 = y 6 = y 10 = y 14 generates the same subgroup N1 = {e, y 2, y 4, y 6 , y 8, y 10, y 12 , y 14 }. Since only the
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elements of order 2 and 4 together will form the subgroup of order 8.Therefore, checking all possible combinations of them,we have N2 = x, y 4 = x, y 12 = {e, x, y 4 , xy 4, y 8 , y 12, xy 8 , xy 12 } ,N3 = xy 2 , y 4 = xy 6 , y 4 = xy 10 , y 4 = xy 2 , y 12 ,N4 = xy 4, y 4 = xy 8, y 4 = xy 12 , y 4,N5 = xy 6, y 4 . Subgroups of order 16: Let K be any arbitrary subgroup of G of order 16 then the elements of K must have order 1, 2,4,8 or 16. If K contains an element of order 16, then K is generated by an element of order 16. The elements of order 16 i.e.,y 3 = y 5 = y 7 = y 9 = y 11 = y 13 = y 15 generates the same subgroup K1 = {e, x, y 2, y 4, y 6 , y 8, y 10, y 12 , y 14 , xy 2, xy 4 , xy 6 , xy 8, xy 10 , xy 12 , xy 14 }. Since only the elements of order 2 and 8 together will form the subgroup of order 16, therefore, checking all possible combinations of them, we have K2 = x, y 2 and K3 = xy, y 2 the subgroups of order 16. G ,,l l , l , l
x, y 2
y
xy, y 2
x, y 8
y 4 xyxy 3 xy 5 xy 7
! ! l ! ,A@ C ! , l ! l , ! CA@ ! l, CA @ !! C A @ xy 6 , y 4xy 4 , y 4 xy 2, y 4 x, y 4 y 2 C A @ X P a ! aX , l ! !!hhh hh aX aQ QP X P C A @ X ! P h l , ! h X P QhX ! h ! ah P X ! h l , ! C A @ a X h P Q ! aQ hP X hP ! h !! l X a hX , C A @
8 xy 6 , yh
xy 4 , y 8
xy 2 , y 8
hhhh ``` aa PP `h aa ah ah !!@ `h ` PP ll ah @ `h ` h aa ! ! a! ` P h `h a ``h P @h l ! aa a P `h `h P aa @ h `h ` l P !! a
,, , , , 14 12 10 8 6 4 2 8 xy xy xy xy xy xy xy x y , hhh ````` X PP HH Q , hhhh `` `` X XXXPP HQ @ `` hh` ` `` h , PPHQ@ ` X h` h ` X h` ` ` H X h ` ` P Q@ , h X ` ` h P H X ` ` h h `h ` X P Q` , ` ` X H h ` ` P h X ` h h , @ X H P Q ` `
{e}
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On subgroups lattice of non-abelian group
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[4] A.A. Gufar and R. Sulaiman, Counting fuzzy subgroups of symmetric groups S2 , S3 and alternating group A4 , J. Quality Measurement and Analysis, 6(1) (2010), 57-63. [5] I.N.Herstein,Topic in algebra, John Wiley and Sons, New York, 1975. [6] J.B.Fraleigh, A first course in abstract algebra, Addison-Wesley, London, 1992. [7] R.Sulaiman, Constructing fuzzy subgroups of symmetric groups S4 , Inter. J. Algebra,6(1)(2012),23-28. [8] V. Murali and B.B.Makamba, Counting the number of fuzzy subgroups of an abelian group of order pn q m , Fuzzy Sets Systems, 144 (2004), 459-470. [9] V.Murali and B.B.Makamba, On an equivalence of fuzzy subgroupsI, Fuzzy Sets and Systems, 123 (2001), 259-264. [10] V. Murali and B.B. Makamba, On an equivalence of fuzzy subgroups II, Fuzzy Sets and Systems, 136(1) (2003), 93-104. [11] A.Rosenfeld, Fuzzy groups, J. Math. Anal. and App., 35 (1971), 512-517. [12] Z.Raza and B.Humera, Fuzzy subgroups of finite abelian group (submitted), Internatonal Journal of Fuzzy Systems,(2012) [13] R.Sulaiman, Subgroups lattice of symmetric group S4 , Inter. J. Algebra, (1) (2012),29-35. [14] M.Tarnauceanu and L.Bentea, On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems, 136 (2003), 93-104. [15] Y.Zhang and K.Zou, A note on an equivalence relation on fuzzy subgroups, Fuzzy Sets and Systems, 95 (1992),243-247. Received: June, 2012
