<<
It is also isomorphic to the homomorphisms on the group Q/Z (i.e.,
fractions with addition modulo 1).
>>
I take it this means Hom(Q/Z, Q/Z) = End(Q/Z).
(If not, what are the homomorphisms into?)
Will just free-associate about this group for a bit, since I've always liked it.
Unlike Q, it's a countable direct sum of subgroups, namely for each prime p
the group of elements
G_p := {k/p^n mod 1 : k in Z, n in Z+, (k, p) = 1 }
under addition.
So a homomorphism h: Q/Z -> Q/Z is determined by its values on all the elements,
for each p, of form 1/p^k, that is, h(1/p), h(1/p^2), . . . , h(1/p^n), . . . with the
requirement, of course, that h(px) = h(x) + . . . h(x) (p summands) for any x and
in particular x = 1/p^n for any n = 1,2,3,....
Say k is the smallest positive integer j such that x= h(1/p^j) is nonzero. Then
clearly px = 0 (mod p), so x= r/p for 0 < r < p.
Then for y = h(1/p^(k+1)) we must have py = x, so y = s/p^2 where ps = r (mod p^2),
so s is any member of the set {r, r+p, . . ., r + (p-1)p} of p possible values. Etc.
So it's beginning to look as though this group End(Q/Z) is the direct sum of the
groups Hom(G_p, G_p) = End(G_p), which are each like the inverse limit
. . . -> Z_p^4 -> Z_p^3 -> Z_p^2 -> Z_p
where each map is multiplication by p.
Okay, that sheds some light on Franklin's comment.
--Dan
