Solved: Do i need to list all the members and check or is it enough, Algebra

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Do i need to list all the members and check or is it enough

Consider the group Z[4] × Z[6] under * such that
(a, b) * (c, d) = (a +[4] c, b +[6] d).
(here +[4] means + is in Z[4] and +[6] is in Z[6])

We would like to find a group of permutations that is isomorphic to Z[4]Z[6].

Is this group cyclic? If so, prove it. If not, explain why.

Do I need to list all the members and check or is it enough to know that Z[2]xZ[12] has the same order. And then check that their identity elements have the same order?

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Algebra: Do i need to list all the members and check or is it enough

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Algebra: Do i need to list all the members and check or is it enough

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