4. The permutations (e, a, B, v) form a group. If e = (1)(2)(3)(4)(5)(6), a = (1)(2)(35)(46), B = (4)(6)(12)(35),find y. Also find the cycle index of this group.

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Answered: 4. The permutations (e, a, B, v) form a…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 25E

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Write down the six elements of S3 and give the multiplication table for the group. The six elements of S3 are then: 2 2 2 3 1 = (1/²/3) a = (2²2 ² 3). b = (² ² 2) 2 3 1 3 1 1 1 2 c = (²2²3). d= (² 2 3). e= (1 ² 3) C 1 3 1 3 The multiplication table for S3 can be written down directly by doing the required composition. Find ab, de,and c^2 1 2 1 2 1 2 2 3 a = (2²2 ²7 ³3) a = ( 2² ²3 ³ ) = b (² ² 2) ²= ( 2₂ ²3 ³) a= 3 1 1 3 1 1. = (1 2² ₂3 ) = 1 2 1 2 2 1 c = ( ²2 ²3 ) c = ( 2₂ ²3 ) = c ² = c = (₁ ²²3) ²¹ C C 1 1 2 3. a3 =
Let a = 1 2 3 4 5 6 7 8 2 3 5 1 4 78 6 " B = 1 3 7 2 3 4 5 6 7 8 8 4 6 5 1 2 (a) Write a, a -1 and ß as product of disjoint cycles. (b) Compute ßaß-¹. Is ßaß-¹ € As? Justify your answer. € S8.

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4. The permutations (e, a, B, v) form a group. If e = (1)(2)(3)(4)(5)(6), a = (1)(2)(35)(46), B= (4)(6)(12)(35), find y. Also find the cycle index of this group.