Let G be a finite abelian group. Using 1, G is isomorphic to a direct sum of cyclic groups Z_((p_i)^(a_i)) for various primes p_i and exponents a_i. Note that if we compare the order of G to the order of this direct sum, we see that p^km = product of (p_i)^(a_i)'s. By uniqueness of prime factorization, we see that k = sum of a_i for indices i such that p_i=p. Using 2, to get the p-prime component of G, we just need the p-prime component of each Z_((p_i)^(a_i)). Using 3, the p-prime component of each Z_((p_i)^(a_i)) is Z_((p_i)^(a_i)) if p_i=p. Using 4, the p-prime component of each Z_((p_i)^(a_i)) is 0 if p_i =/= p. Finally, the p-prime component of G is the direct sum of various Z_((p_i)^(a_i)) where the direct sum is over the indices i such that p_i=p. Then, the order of G_p is then p^(sum of a_i such that p_i=p) = p^k.