The Conjugacy Classes of Some Three-Generator Groups

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Menemui Matematik (Discovering Mathematics) Vol. 38, No. 1: 22-26 (2016)

The Conjugacy Classes of Some Three-Generator Groups

Alia Husna Mohd Noor1, Nor Haniza Sarmin2 and Sanaa Mohamed Saleh Omer3

1, 2 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor, Malaysia.

3 Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya.

1mylife_syafia@yahoo.com, 2nhs@utm.my, 3mohamedsana51@yahoo.com

ABSTRACT

The conjugacy class of an element in a group is the set of all conjugates of that element in the group. An element x is conjugate to y in a group G if there exists an element g in G such that = y. Many of previous researchers only found the lower bound of the conjugacy classes without giving the exact number of conjugacy classes. In this paper, the number of the conjugacy classes of 3-generator groups of order 16 is computed.

Keywords: Conjugacy class; Groups of order 16.

INTRODUCTION

The conjugacy class of a group is sometimes viewed as a partition of a group. Let a be an element of a group G. The conjugacy class of the element a is defined by the following; Let b be an element in G. Then a and b are conjugate if = for some x in G. The conjugacy class of a is written as cl(a) = { |xϵ G}. In recent years, a number of researchers have studied on the bounds on the number of conjugacy classes. For any non-trivial group G, the number of conjugacy classes, cl > 1. In 1968, Erdos and Turan stated that the number of conjugacy classes can be calculated using the formula cl > | |. In the same year, Polan (1968) has made an improvement on the bound of cl for nilpotent group which is cl > . Besides, Sherman (1979) also investigated the best lower bound for the conjugacy classes. As for the case that if G is a nilpotent group, Sherman revealed that cl > c(| | / − 1) + 1 where c is the class of nilpotent group.

However, the improvement made by Sherman (1979) was still weak. He gave a better estimation on the bounds of the conjugacy classes; however it is still far away from the exact value. In 2008, Ahmad found the exact number of conjugacy classes for 2-generator p-groups of class 2. The scope of this paper is on the 3-generator groups of order 16. There are three groups under this category, , , and given by the following presentations (S.Ok, 2001):

= <x, y, z | = = = 1, [x,z] = [y,z] = 1, [x,y] = >,

= <x, y, z | = = = 1, [x,z] = [y,z] = 1, = >,

= <x, y, z | = = = 1, [x, y] = z, [x,z] = [y, z] = 1 >.

This paper is structured as follows: The first section is the introduction section. It includes basic definitions and concepts in group theory that are used in this research.

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The Conjugacy Classes of Some Three-Generator Groups

Menemui Matematik Vol. 38 (1) 2016 23 PRELIMINARIES

In this section, some basic properties which are needed later are presented.

Proposition 1.1 (Fraleigh, 2000) The conjugacy class of the identity element is its own class, namely cl(1) = {1}.

Proposition 1.2 (Fraleigh, 2000) Let a and b be two elements in a finite group G. The elements a and b are conjugate if they belong in one conjugacy class, that is cl(a) = cl(b).

Proposition 1.3 (Fraleigh, 2000) Suppose a is an element of a group G, then a lies in the center Z(G) if and only if its conjugacy class has only one element.

MAIN RESULTS

In this section, the number of conjugacy classes of all three generator groups of order 16 are computed and determined. The number of conjugacy classes of a group G is denoted by K(G).

Theorem 2.1 Let = <x, y, z | = = = 1, [x, z] = [y, z] = 1, [x, y] = >. Then the number of conjugacy classes of , K ( ) = 10.

Proof: Using definition of conjugacy class, the conjugacy classes of can be found. Based on the definition, cl(a) = { ϵ : there exists x ϵ with = }. Thus, the conjugacy classes of are determined as follows: First, let a = 1. Since the conjugacy class of the identity element is the identity itself, then, cl(1) = {1}.

Next, we find the conjugacy class for a, cl(a) = { a } where ϵ {1, x, y, z, , , xy, xz, , , yz, , , xyz, , } for all a in . The conjugate elements of a are determined as follows:

When = x, a = (x)x( ) = xxx = x,

when = y, a = (y)x( ) = yxy = yy = , when = z, a = (z)x( ) = zxz = zx = xz = x,

when = , a = ( ) ( ) = = zzxzz = zxzzz = xzzzz = x, when = , a = ( ) ( ) = = zx = x = x,

when = xy, a = (xy) ( ) = ( ) ( ) = xyxxy = xyy = x , when = xz, a = (xz) ( ) = ( ) ( ) = xz = x,

when = x , a = (x ) ( ) = ( ) ( ) = x = x,

when = x , a = (x ) ( ) = ( ) ( ) = x = x,

when = yz, a = (yz) ( ) = ( ) ( ) = yzxy = yzyx = yyzxz = xzz = x ,

when = y , a = (y ) ( ) = ( ) ( ) = y = y = y yx =

yzzyx = zzx = = zzx = zxz = xzz = x ,

when g = y , a = (y ) ( ) = ( ) ( ) = y = y yx = yzzzyx z = yzzyzx = yzzyzx = yzyzzx = zzzx = xx = x ,

when = xyz, a = (xyz) ( ) = (x ) ( ) = xyzyz = xzz = x ,

when = xy , a = (xy ) ( ) = (x ) ( ) = xy y = xyzzy = xzyzy = xzz = x , when = xy , a = (xy ) ( ) = (x ) ( ) = xy y = xy zy = xy y = xyzzy = xzyzy = xzz = x .

Thus, cl(x) = {x, x } . Since x and x belong in one conjugacy class, then, cl(x) = cl(x ).

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Alia Husna Mohd Noor, Nor Haniza Sarmin and Sanaa Mohamed Saleh Omer

Menemui Matematik Vol. 38 (1) 2016 24 By replacing with the rest of the elements in the set { y, z, , , xy, xz, , , yz, ,

, xyz, , }, the conjugacy classes of are found as follows:

(i) cl(1) = {1},

(ii) cl(x) = {x, x } = cl(x ), (iii) cl(y) = {y, yz } = cl(y ), (iv) cl(z) ={ z},

(v) cl( ) ={ }, (vi) cl( ) = { },

(vii) cl(xy) ={ , } = cl(xy ), (viii) cl(xz) ={ , } = cl(x ), (ix) cl(yz) ={ , } = cl(y ), (x) cl(xyz) ={ , } = cl(xy ).

Therefore, K( ) = 10.

Theorem 2.2 Let = <x, y, z | = = = 1, [x, z] = [y, z] = 1, = >. Then the number of conjugacy class of , K ( ) = 10.

Proof: The conjugacy classes of are computed in a similar way for . The conjugacy classes of are calculated as follows:

For all a in , cl(a) = { a } where ϵ {1, x, y, z, , , xy, xz, , , yz, , , xyz, , }. The conjugate elements of a are determined as follows:

When = x, a = (x)x( ) = xx = x, when = , a = ( ) x( ) = x = x ,

when = , a = ( ) x( ) = = ,

when = y, a = (y)x( ) = yxy = yy = , when = z, a = (z) ( ) = zxz = zzx = x,

when = xy, a = (xy) ( ) =xyxxy = y xxy = yxy = yy = ,

when = y, a = ( ) ( ) = xxyxxxy = xy = xyxxy = y xxy = yxy = yy

= ,

when = y, a = ( y) ( ) = ( ) ( ) = yx y = ,

when = xz, a = (x ) ( ) = xzx z = x ,

when = z, a = ( z) ( ) = xxzxxxz = xzxxxxz = x, when = z, a = ( z) ( ) = ( ) ( ) = xzzx = x,

when = yz, a = (yz) ( ) = ( ) ( ) = zyxyz = zyy z = zxxxz = xzxzx = xzzxx = , when = xyz, a = (xyz) ( ) = xyzxxyz = xyxzxyz = y xzxyz = yzxyz =zyy z = zxxxz

= xzxzx = xzzxx = ,

when = yz, a = ( yz) ( ) = = = xzxxyz =

xyxzxyz = y xzxyz = yzxyz = zyy z = zxxxz = xzxzx = xxzzx = ,

when = y , a = ( y ) ( ) = = .

Since cl(x) = {x , }, then cl(x) = cl( ).

The same steps are used for = {y, z, , , xy, xz, , , yz, , , xyz, , }.

Therefore, the conjugacy classes of are listed as follows:

(i) cl(1) = {1},

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The Conjugacy Classes of Some Three-Generator Groups

Menemui Matematik Vol. 38 (1) 2016 25 (ii) cl(x) = {x, } = cl( ),

(iii) cl( ) = { },

(iv) cl(xy) = {xy, } = cl( ), (v) cl(xz) = {xz, } = cl( ), (vi) cl(y) = {y, } = cl( ), (vii) cl(z) ={z},

(viii) cl( ) = { },

(ix) cl(yz) = {yz, } = cl( ), (x) cl(xyz) = {xyz, } = cl( ).

Thus, K( ) = 10.

Theorem 2.3 Let = <x, y, z | = = = 1, [x, y] = z, [x, z] = [y, z] = 1 >. Then the number of conjugacy class of , K ( ) = 10.

Proof: By using the same steps as in the Theorem 2.1 and Theorem 2.2, the conjugacy classes of are obtained. Therefore, the conjugacy classes of are computed as follows:

Using the definition, cl(a) = { a } where ϵ {1, x, y, z, , , xy, xz, , , yz, , , xyz, , }. The conjugate elements of a are determined as follows:

When = x, a = (x)x( ) = xx = x, when = , a = ( )x( ) = x = x,

when = , a = ( )x( ) = = ,

when = y, a = (y)x( ) = yxy = yyxz =xz, when = z, a = (z) ( ) = zxz = xzz = x,

when = xy, a = (xy) ( ) =xyx( yz) = xyx yz = xyyz = xz,

when = y, a = ( ) ( ) = yx y = xxyxxxy = xyxzxxxy = xyzxxxxy = xyzy = xzyy = xz,

when = y, a = ( y) ( ) = ( ) ( ) = yxxyz = xyxxyz = yxzxxyz = xxyzxxxyz = xyxzzxxxyz = xz,

when = xz, a = (x ) ( ) = xzx z = x,

when = z, a = ( z) ( ) = zx z = xxzx z= xzxx z = x, when = z, a = ( z) ( ) = ( ) ( ) = xzxz = zxz = xzz = x, when = yz, a = (yz) ( ) = ( ) ( ) = yzyxzz = yzyx = zyyx = xz, when = xyz, a = (xyz) ( ) = xyzx y = xyzy = xyyz = xz,

when = yz, a = ( yz) ( ) = = = zzx yz = xz,

when = y , a = ( y ) ( ) = = = =

= = = = = .

Since cl(x) = {x, }, then cl(x) = cl( ).

The calculations are continued with = {y, z, , , xy, xz, , , yz, , , xyz, , }. The conjugacy classes of are listed in the following:

(i) cl(1) = {1},

(ii) cl(x) = {x, } = cl( ), (iii) cl( ) = { },

(iv) cl(xy) = {xy, } = cl( ), (v) cl(y) = {y, } = cl( ),

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Alia Husna Mohd Noor, Nor Haniza Sarmin and Sanaa Mohamed Saleh Omer

Menemui Matematik Vol. 38 (1) 2016 26 (vi) cl(z) ={z},

(vii) cl(xy) = {xy, } = cl( ), (viii) cl( ) = { , } = cl( ),

(ix) cl( ) = { , } = cl( ),

(x) cl( ) = { }.

Hence, K( ) = 10.

CONCLUSION

Based on the calculation in Theorem 2.1, 2.2 and 2.3, it can be seen that the number of conjugacy classes for all three-generator groups of order 16 are equal to 10. However, the proof for each group is different due to the different relations given in the group presentation.

REFERENCES

Ahmad, A. The Exact Number of Conjugacy Classes for 2-generator p-groups of Nilpotency Class 2. Ph.D Thesis. Universiti Teknologi Malaysia; 2008.

Bertam, E. A. Herzog, M and Mann, A. (1990), On Graph Related to Conjugacy Classes of Groups. Bull. London Math. Soc., 22: 569 – 575.

Bianchi, M., Camina, R. D., Herzog, M., and Pacifici, E. (2015), Conjugacy Classes of Finite Groups and Graph Regularity. Forum Mathematicum 27(6): 3167 – 3172.

Erdos, P. and Turan, P. (1968), On Some Problems of Statistical Group-Theory. IV. Acta Math.

Hung. 19(3-4): 413 – 435.

Fraleigh, J.B. (2000), A First Course in Abstract Algebra, 7th ed. USA: Addison Wesly Longman, Inc.

Moreto, A., Qian, G. and Shi, W. (2005), Finite Groups whose Conjugacy Class Graphs Have Few Vertices. Arc. Math., 85(2): 101 – 107.

S. Ok, K. (2001), On p-groups of Order p4. Comm. Korean Math. Soc., 16(2): 205 – 210.

Poland, J. (1968), Two Problems on Finite Groups with k Conjugate Classes. J. Austral. Math.

Soc., 8(01): 49 – 55.

Sherman, G.J. (1979), A Lower Bound for the Number of Conjugacy Classes in a Finite Nilpotent Group. Pac. J. Math., 80(1): 253 – 254.

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