Q7 Let U(18) be the set of all positive integers less than 18 and relativelyprime to 18. Find the order of the group (U(18), ®1s).Q8 Let S(2, R) = {[a b]: a, b, c, d € R and ad – be #0. Find the order0 -1the group S(2, R) under matrix multiplication.of the elementQ 9 Find the divisors of zero in (Z12, ®12, O12).

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Answered: Q7 Let U(18) be the set of all positive…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.6: Matrices

Problem 24E

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The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of section 3.1. Find the order of each element of the group. (Sec. 3.1,35) A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.
Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.
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Q7 Let U(18) be the set of all positive integers less than 18 and relatively prime to 18. Find the order of the group (U(18), 1s).