1. 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) matrices1001ω 0010 ω0ω 000{(69) -(69) (69) (18)-(25)·(25)}under matrix multiplication, where w = C and w³cube root of unity.)(iii) ZX={1, 2, 4, 5, 7, 8} under multiplication modulo 9.(iv) Z2 × Z3.=1,w 1. (That is, w is a particular

Question 1:Step 1:Step 2:Step 3:

Answered: 1. Find which pairs of the following…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.2: Complex Numbers And Quaternions

Problem 50E

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Q1: Check whether (K,*) forms a group, where "* "refers to the matrix multiplication and K = {h₁, h₂, h3, h4}. Represent the properties in table form. Here, m₁ = ( 9 ), h₂ = ( 1), h3 = (₁-¹) and h₁ = (¹9).
1.)Integrate: ∫(x+1)(x+2)^6dx Group of answer choices a.)(2/56) (7x + 6)(x + 2)7 + C b.)(2/56) (7x - 6)(x + 2)7 + C c.)(1/56) (7x - 6)(x + 2)7 + C d.)(1/56) (7x + 6)(x + 2)7 + C 2.)Find the derivative of y = log26x. Group of answer choices a.)3.44 b.)1.44 c.)2.44 d.)0.44
An operator T on a complex inner product space is normal if TT - TT. State the Complex Spectral Theorem b) Which of the following matrices can be diagonalized over C²? (14) (3-3)
Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In general, we apply the row element on the left of the column element.)
Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D %3D where a = (1 2 3 4) and b = (2 4). Compute (ab)(ab) Compute (ab)(a²b) Compute (ab)(a³b)
Solve 2b and 2a
a) Show that 0 = {( 96 ) ( ) ( ¯)} P {(9), C has an identity element with respect to matrix multiplication. b) Does each element of P have an inverse with respect to matrix multiplication? c) Is P a group with respect to matrix multiplication?
2. Frobenius emphasized in his research how matrices could be used to mimic other mathematical systems. Here, we mimic the behaviour of complex numbers using matrices. Mathematicians call such a relationship an isomorphism. Complex number - Matrix 9. a+bi 9- a Note that the complex number can be read off the top line of the matrix. Thus, 21 4 and -2 2+3/ 4-21. 4 a) Find the matrices corresponding to 2-5i and 1+37. b) Multiply the two matrices. c) Find the corresponding complex number for the matrix found in part (b).
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Which of the following are the subgroups? a) The set of all nxn matrices with determinant 2. b) The set of all upper triangular nxn matrices with no zeros on the diagonal. c) The set of all nxn matrices with determinant-1. d) All the above. Answer OA Oc
Abstract Algebra

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