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Math Help - Cyclic Group

  1. #1
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    Cyclic Group

    Let G={ f|f:\mathbb{N}\rightarrow\mathbb{Z}_2} and clearly G is a group under usual addition of functions. Show that each eleemnt of G is of finite order. Moreover, show that G is a group of infinite order.

    I have no idea where to start, can anyone give me some tips? Thanks
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  2. #2
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    Re: Cyclic Group

    here is a hint: we can regard each f as a subset of N, for any f, we consider the subset of N: {n in N: f(n) = 1}. show that for f,g in G, we can regard f+g as the symmetric difference of the respective associated subsets. conclude that f+f = 0 for every f in G.

    to show G is infinite, consider for each k in N, the element of G which we can denote fk:

    fk(n) = 1, if n = k
    fk(n) = 0, if n ≠ k

    i'm not sure why this thread is titled "cyclic group", as the group in question is not cyclic.
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