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What is the minimum number of generators needed for Z2 ⊕ Z2 ⊕Z2? Find a set of generators and relations for this group.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 21E: Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and...

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What is the minimum number of generators needed for Z₂ ⊕ Z₂ ⊕
Z₂? Find a set of generators and relations for this group.

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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 =
Can you prove that a set is a group, without having an operation? for example can you prove this set B={(1+2A):(1+2N)} is a group? Is the operation division in this case or is the exercise wrongly given?
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