Let { } 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
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.