Consider the set Z, × Z, = {(x, y) : x € Z, and y € 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) (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. 1. a. List the element of Z, x Z3. b. Determine the identity element in Z, × Z3. c. Determine the inverse element of each element in Z, x Z3. d. Is Z, x Zz a group under the given operation? Why? Is it Abelian? e. List a proper subgroup of Z, × Z3. Write "none" if there are no proper subgroups.

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