Question 1.State whether True or False. Provide a reason in each case.(a) The pair (Z321,-) that consists of the set of congruence classes modulo 321 together withmultiplication, has two zero divisors.(b) Every integral domain consists of an Abelian additive group, and a commutative unitalring which has a unity, but no zero divisors.(c) The set nz for n N, the set R\Q, and the set C\R have no zero divisors.(d) Given the set Z24 of congruence classes modulo 24. Then the congruence classes [2], [3], [4],[6], [9], [12], [18], and [20] are some of the zeros divisors of Z24.(e) The triple (Z[√-3], +,-) consisting of the set of algebraic numbersZ[√-3] = {+ + √-3: for k, l, m, n € Z and I, n ‡ 0}with operations of addition + and multiplication, constitutes an integral domain.(f) The set of all functions M ([0,1]) from the the closed unit interval [0, 1] onto [0, 1],together with usual function addition and function multiplication, does not have any zerodivisors. If you think it does, then give an example.

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Answered: Question 1. State whether True or…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 27E

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Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .
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25. Prove or disprove that every group of order is abelian.

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