3. Which of the following groups are cyclic?(i) The group G of positive rational numbers which can be written as a/b with a and b odd,under multiplication.(ii) Z{1, 2, 4, 5, 7, 8} under x modulo 9.(iii) The group H of matrices of the form=1 n011with n Z, under matrix multiplication.

A cyclic group is a group that can be generated by a single element.This means there exists an…

Answered: . Which of the following groups are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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- Find which pairs of the following groups are isomorphic: (i) The dihedral group D3 of symmetries of an equilateral triangle. (ii) The group of (complex) matrices 10 W 0 0 01 0 2 { ( 1 ; ). ( 2 ). ( ² ). ( i ). ( ² ). ( ² )} 1 0 w2 0 W 10 W 0 2 0 = under matrix multiplication, where we C and ³ cube root of unity.) (iii) Zỗ = {1,2, 4, 5, 7, 8} under multiplication modulo 9. (iv) Z2 × Z3. 1,w1. (That is, w is a particular
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. Which of the following groups are cyclic? (i) The group G of positive rational numbers which can be written as a/b with a and b odd, under multiplication. (ii) Z = {1, 2, 4, 5, 7, 8} under x modulo 9. (iii) The group H of matrices of the form (17), 01 with n Z, under matrix multiplication.