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    • Thread Starter

    Hi, I have the following:

    Let \Omega be a discrete subgroup of C, the complex plane.

    If: i) \Omega = {nw_1 | n \in Z} , then \Omega is isomorphic to Z.

    ii) \Omega = {nw_1 + mw_2 | m,n \in Z} where w_1/w_2 \notin R , then \Omega is isomorphic to Z x Z

    So from what I understand isomorphic is a map that is one to one between two sets that preserves the binary relatione exisising between elements, that is f(x*y)=f(x)*f(y) (1), where * is the operation the map is isomorphic to.

    So to define a isomorphism you need to define: - two sets - the map between them - the relevant operation which is preserved, defined by (1)

    So, my book doens't say which operation, is it addition, it also doesn't say which map - is the map to take the integer with the map f = n in case i) and f=n+m in case 2, under the operation addition it is then easy to show that (1) is obeyed in both cases?

    By the wording it seems to imply the fact that w_1/w_2 \notin R is significant for there to be an isomorphism to Z x Z, I don't at all understand why, can someone explain? Many thanks in advance
    • Thread Starter

    apologies i now know the answer.
    cant delete your own threads?
    could a moderator close it or?
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