ORIGINAL ARTICLE Commutator properties and relative theories in communication with solvable groups Md Taufiq Nasseef Md Taufiq Nasseef. Commutator properties and relative theories in branches of Mathematics, has developed in many different directions. It becomes communication with solvable groups. J Pur Appl Math. 2018;2(2):14-21. of prime importance in many Mathematical disciplines. Here, I have studied an essential part of groups called Commutators related to solvable groups. In this ABSTRACT study, I have discussed Commutator properties and intend to present some relative The study of groups arose from the theory equations; more specifically form the noteworthy theorems of commutators in connection with solvable groups. I have attempt to find roots of a polynomial in terms of its coefficients, early in the 19th assumed the group G to be finite group throughout the whole work. century. The theory of groups itself, which had already been applied in almost all Key Words: Finite group; Commutator; Commutator subgroup; Solvable groups Definition of a commutator Let x,y,z ∈G be elements of a group G then (a) Let G be a group. If a and b are elements of a group G, then ab=bac for (xy,z) = (x,z)y(y,z) = (x,z)(x,z,y)(y,z) [4] some c∈G. If a and b commute, then, of course, c=e. In general c ≠ e and c −− Proof: = a11 b ab . An element of this form is called a commutator and is usually denoted by (a,b). Consider, (xy,z) = (xy)-1z-1(xy)z = y-1x-1z-1xyz (b) We also define commutators of higher order by the recursive rule: Consider, (x,z)y(y,z) = y-1(x,z)y(y,z) −−11 = y-1x-1z-1xzyy-1z-1yz ( xx12,, xxnn−− 1 ,) = (( xx12 ,, xn 1) , x n) =( Xx , n) = XxXx nn [1] = y-1x-1z-1xyz Where Again, (x,z)(x,z,y)(y,z) These are called simple commutators. = x-1z-1xz(x,z)-1y-1(x,z)yy-1z-1yz (c) The set of all elements which can be obtained by successive commutation = x-1z-1xzz-1x-1zxy-1x-1z-1xzyy-1z-1yz are called complex commutators. = y-1x-1z-1xyz (d) Let G be a group. Let A and B be subgroups by the notation (A,B), we Hence, (xy,z) = (x,z)y(y,z) = (x,z)(x,z,y)(y,z) mean the group generated by all commutators. (a,b) with a∈A, b∈B. Property-3:Let x,y,z be elements of a group G. Then (e) If A1, A2,……An-1 , An are subgroups of a group G, then we define z (A1,A2,……An-1,An) = ((A1,A2,……An-1),An) [2] (x,yz)=(x,z)(x,y) =(x,z)(x,y)(x,y,z) [5] (f) We shall represent conjugation by an exponent ax=x-1ax where x is fixed Proof: in G and for all x∈G. [1] Consider, (x,yz) = x-1(yz)-1xyz = x-1z-1y-1xyz Definition of commutator subgroup Consider, (x,z)(x,y)z = (x,z)z-1(x,y)z Let G be a group. Let us denote by the subgroup generated by the set of all -1 -1 -1 -1 -1 commutators (a,b )= a-1b-1 of G, for all a,b ∈G, then is called the commutator = x z xzz x y xyz subgroup of G′ [1, 7-11]. = x-1z-1y-1xyz Note: G′ is normal in G. Again (x,z)(x,y)(x,y,z) = x-1z-1xzx-1y-1xyx-1(y,z)-1x(y,z) =x-1z-1xzx-1y-1xyy-1x-1yxz-1x-1y-1xyz Some properties of commutators = x-1z-1y-1xyz Note: Let x,y ∈G then (x,y)=e the identity of G if and only if xy = yx, the proof follows directly from the definition of a commutation [1-11]. Hence, (x,yz) = (x,z)(x,y)z = (x,z)(x,y)(x,y,z) Property-1:Let xy∈G be elements of a group G, the (y,x) = (x,y)-1 [3] Property-4: Proof: Let x,y,z be elements of a group G, Then L.H.S: (y,x) = y-1x-1yx = ( x-1y-1xy)-1 = y-1x-1yx (x,y-1,z)y(y,z-1,x)z(z,x-1,y)x = e, the identity of G. [6] Hence, (y,x) = (x,y)-1 Proof: Property-2: Consider (x,y-1,z)y = y-1(x,y-1,z)y Douglas Mental Health University Institute, 6875 Lasalle Blvd. H4H 1R3, Montreal, Quebec, Canada. Correspondence: Md Taufiq Nasseef, Douglas Mental Health University Institute, 6875 Lasalle Blvd. H4H 1R3, Montreal, Quebec, Canada. Tel: 5148626907; email: [email protected] Received: July 18, 2018, Accepted: August 17, 2018, Published: August 23, 2018 This open-access article is distributed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC) (http:// creativecommons.org/licenses/by-nc/4.0/), which permits reuse, distribution and reproduction of the article, provided that the original work is properly cited and the reuse is restricted to noncommercial purposes. For commercial reuse, contact [email protected] J Pur Appl Math Vol 2 No 2 August 2018 18 Nasseef = y-1(x,y-1)-1z-1(x,y-1)zy Property-7: = y-1yx-1yxz-1x-1yxy-1zy Let x,y be elements of a group G and suppose that z = (x,y) commutes with both x and y = x-1yxz-1x-1y xy-1zy (a) Then (xi,yj) = zij for all j [9] Similarly, (y,z-1,x)z = z-1(y,z-1,x)z (i) 1 i ii(− 1) = z-1(y,z-1)-1x-1(y,z-1)xz (ii) ( yx) = z2 yii x for all I [10] = z-1zy-1z-1yx-1y-1zyz-1xz Proof: =y-1z-1yx-1y-1zyz-1xz (b) Again, -1 -1 -1 -1 i -1 i i i i (z,x-1,y)x = x-1(z,x-1,y)x Since x y xy = z, then we have y xy = xz where y x y = (y xy) = (xz) = x z as x and z commute. Conjugating by y which gives that = x-1(z,x-1)-1y-1(z,x-1)yx y-1(y-1xiy)y = y-1 (xizi)y = (xizi)zi = xiz2i = x-1xz-1x-1zy-1z-1xzx-1yx as y and z commute. =z-1x-1zy-1z-1xzx-1yx (c) Repeating this argument j times. We conclude that Now, Combining (a),(b),(c), we get y-jxiyj = xizij and (xi,yj) = zij x-1y-1xz-1x-1yxy-1zy y-1z-1yx-1y-1zyz-1xz z-1x-1zy-1z-1xzx-1yx =e(identity) Again, (ii) holds, for j=1, assuming the result for (i-1), we have -1 y -1 z -1 x Hence, (x,y ,z) (y,z ,x) (z,x ,y) = e, the identity of G. 1 i i-1 (ii−− 1)( 2) (yx) = (yx) (yx) = z2 yii−−11 x yx i Property-5: Since, by then follows at once, that is, Let x,y,z be elements of a group G, then x y x y 1 (x,y,z)(y,z,x)(z,x,y)=(y,x)(z,x)(z,y) (x,y)(x,z) (y,z) (x,z)(z,x) [7] i ii(− 1) Hence ( yx) = z2 yii x for all i Proof: L.H.S. (x,y,z)(y,z,x)(z,x,y) Property-8: = (x,y)-1z-1(x,y)z(y,z)-1x-1(y,z)x(z,x)-1y-1(z,x)y Let x,y,z be element of a group G then = y-1x-1yxz-1x-1y-1xyzz-1y-1zyx-1y-1z-1yzxx-1z-1xzy-1z-1x-1zxy (i) If is commutes with z and if (x,G) is abelian then = y-1x-1yxz-1x-1y-1xzyx-1y-1z-1yxzy-1z-1x-1zxy (x,y,z)=(x,z,y) [11] R.H.S. (y,x)(z,x)(z,y)x(x,y)(x,z)y(y,z)x(x,z)(z,x)y (ii) If (x,y) commutes with both x and y then = (y,x)(z,x)x-1(z,y)x(x,y)y-1(x,z)yx-1(y,z)x(x,z)y-1(z,x)y (xy-1)-1=(x-1,y)=(x,y-1) [12] = y-1x-1yxz-1x-1zxx-1z-1y-1zyxx-1y-1xyy-1x-1z-1xzyx-1y-1z-1yzxx-1z-1xzy-1z-1x-1zxy Proof: = y-1x-1yxz-1x-1y-1xzyx-1y-1z-1yxzy-1z-1x-1zxy First (x,y,z ) = (x,y)-1z-1(x,y)(z) Therefore, L.H.S. = R.H.S = y-1x-1yxz-1x-1y-1xyz Property-6: = x-1(xy-1x-1y)(xz-1x-1z)(z-1y-1xyz) Let x,y,z be elements of a group G, then Furthermore, xy-1xy = (x-1,y) = (xm,y) for some positive integer m, we conclude easily by p-2 that (xm,y)∈(x,G) . (x,y,z) = ((z,x-1,y-1)xy)-1((y-1,z-1,x)zy)-1 [8] Thus, xy-1x-1y, and likewise xz-1x-1z lies in (x,G). Proof: Hence, by hypothesis these two elements commute. It follows, therefore, -1 -1 L.H.S (x,y,z) = (x,y) z (x,y)z that -1 -1 -1 -1 -1 = y x yxz x y xyz (x,y,z) = x-1(xy-1x-1y)(xz-1x-1z)(z-1y-1xyz) -1 -1 xy -1 -1 -1 -1 -1 R.H.S.