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 0 0 0 0 2 0 { ( 1 ) . ( 8 ) . ( ² ). ( ; ; ). ( ² ). ( ²6 ) } №²2 0 w² 0 W 1 W 0 under matrix multiplication, where we C and w³ cube root of unity.) {1, 2, 4, 5, 7, 8} under multiplication modulo 9. (iii) Z (iv) Z2 × Z3. = = 1,w1. (That is, w is a particular

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
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Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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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
Transcribed Image Text:- 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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