Problem 1. In each of the following, either prove or disprove that the given pair(G,) is a group.1.1. G = R with binary operation ⚫ defined byab=a+b+ab Va, bЄ R.1.2. G is the space of matricesG = {A Є M2(Z) : det(A) = 0}with operation ● given by matrix multiplication.1.3. G is the set of matricesG= {A Є M2(Z20) : det(A) = 1}with operation ⚫ given by matrix multiplication. Here Zo denotes the set ofall non-negative integers.

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Answered: Problem 1. In each of the following,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 5E: The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of...

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11. Prove that the set of all 2 × 2 matrices with entries from R and de- terminant +1 is a group under matrix multiplication.
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