Please answer letter D and E onlyyy. 1.Consider the set Z, × Z3 = {(x, y) : x E Z, and y e Z3}. Define theoperation/sum O in Z2 × Z3 such that for any two elements (a, b) and (c, d)in Z, x Z3, we have (a,b) O (c, d) = (a +2 c, b +3 d) where +2 is theaddition modulo 2 in Z2, and +2 is the addition modulo 3 in Z3.a. List the element of Z, × Z3.b. Determine the identity element in Z, × Z3.c. Determine the inverse element of each element in Z, × Z3.d. Is Z, x Z, a group under the given operation? Why? Is it Abelian?e. List a proper subgroup of Z, × Zz. Write "none" if there are no propersubgroups.

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Answered: Please answer letter D and E onlyyy.

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Advanced Math

Please answer letter D and E onlyyy.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...

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Question

Please answer letter D and E onlyyy.

1.
Consider the set Z, × Z3 = {(x, y) : x E Z, and y e Z3}. Define the
operation/sum O in Z2 × Z3 such that for any two elements (a, b) and (c, d)
in Z, x Z3, we have (a,b) O (c, d) = (a +2 c, b +3 d) where +2 is the
addition modulo 2 in Z2, and +2 is the addition modulo 3 in Z3.
a. List the element of Z, × Z3.
b. Determine the identity element in Z, × Z3.
c. Determine the inverse element of each element in Z, × Z3.
d. Is Z, x Z, a group under the given operation? Why? Is it Abelian?
e. List a proper subgroup of Z, × Zz. Write "none" if there are no proper
subgroups.

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