by the structure theorem for finitely-generated abelian groups we may write:
, for primes p1,...,pk.
since G is assumed non-cyclic, we cannot have gcd(p1,...,pr) = 1 (by the chinese remainder theorem), so for some i ≠ j, pi = pj.
without loss of generality (we may re-arrange the direct sum as we like) take i = 1, j = 2.
now has a subgroup isomorphic to
(since each is cyclic and thus has a cyclic subgroup of order pi).
but p1 = p2, so setting p = p1 = p2:
is isomorphic to a subgroup of G.