GRE Subject Test Question:
Up to isomorphism, how many abelian groups are there of order 36?
The answer given is 4 and the explanation is as follows: Let G be an abelian group with order n. Then G is isomorphic to the products of the form $$Z_{(p_1^{n_1})} \times Z_{(p_2^{n_2})} \times \cdots \times Z_{(p_k^{n_k})}$$ where the $p_j$'s are not necessarily distinct, are the primes in the factorization of n and that $(p_1^{n_1})(p_2^{n_2}) \cdots (p_k^{n_k})=n$. Here $Z_n$ denotes the cyclic group of $\{0,1,2,3,4,5,6,7\}$ under addition modulo n. For $n = 36 = 2^2 3^2$, G is isomorphic to $$Z_{(2^2)} \times Z_{(3^2)} \\ Z_{(2^1)} \times Z_{(2^1)} \times Z_{(3^2)} \\ Z_{(2^2)} \times Z_{(3^1)} \times Z_{(3^1)} \\ Z_{(2^1)} \times Z_{(2^1)} \times Z_{(3^1)} \times Z_{(3^1)}$$
I have two questions:
- Could someone provide me with a link to a proof of the statement that G is isomorphic to the products of the form $Z_{(p_1^{n_1})} \times Z_{(p_2^{n_2})} \times \cdots \times Z_{(p_k^{n_k})}$ ?
- Why does $Z_n$ denote the cyclic group of $\{0,1,2,3,4,5,6,7\}$ under addition modulo n?
Thanks in advance and sorry if this is a basic question. I'm a physics major trying to make the transition to applied mathematics in graduate school and I'm afraid my abstract mathematics are rather weak!