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 f_{k}:

f_{k}(n) = 1, if n = k

f_{k}(n) = 0, if n ≠ k

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