Fundamental Theorem of Finite Abelian Groups

Question

Fundamental Theorem of Finite Abelian Groups indicates that $\mathbb{Z}_{n}$ is isomorphic to $\mathbb{Z}_{p_1^{k_1}} \times \mathbb{Z}_{p_2^{k_2}}\times$ ... $\times\mathbb{Z}_{p_n^{k_n}}$ where $p_i$ are prime and not necessarily distinct.

But I know that $\mathbb{Z}_4$ is not isomorphic to $\mathbb{Z}_2 \times\mathbb{Z}_2$ because $\mathbb{Z}_{mn}$ is only isomorphic to $\mathbb{Z}_m\times\mathbb{Z}_n$ only if $m$ and $n$ are relatively prime.

Can anyone clarify and correct my misunderstanding on this subject.

Thanks

Answer 1

Given a finite Abelian group $G$, $G$ is isomorphic to a direct product of cyclic groups: $Z_{{p_1}^{e_1}}\times Z_{{p_2}^{e_2}}\times Z_{{p_3}^{e_3}}\times ...Z_{{p_n}^{e_n}}.$ But these primes may not be distinct.

It seems that your misunderstanding stems from thinking that any finite Abelian group must decompose entirely into a product of cyclic groups whose orders are powers of distinct primes.

Answer 2

There is nothing wrong with the statement. You have to note that $\mathbb{Z}_4\cong\mathbb{Z}_{2^2}$ which agrees with the statement.

Your misunderstanding may have arisen from thinking that the factorisation of the group follows the factorisation of the order of the group, $n$ in your case. So you would think that $\mathbb{Z}_4\cong\mathbb{Z}_2\times\mathbb{Z}_2$ which is wrong.

Hope this helps.