Determine abelian groups with 48 elements

Question

I am just doing some revision for my linear algebra exam, and I came across this problem:

Determine all abelian groups (up to isomorphism) with exactly 48 elements.

I am not sure I have ever seen a similar problem, so I am not sure how to approach it. Does anybody know how to solve this?

Answer 1

The most important theorem (maybe) for Abelian groups is Fundamental Theorem of Finitely Generated Abelian group :

For any finitely generated abelian group $G$, it is isomorphic to \begin{align} \mathbb{Z}^{r}\times \mathbb{Z}/p_{1}^{e_{1}}\times\cdots\times\mathbb{Z}/p_{r}^{e_{r}} \end{align} for some $r, e_{1}, \dots, e_{r}\geq 0$ and prime numbers $p_{1}, \dots, p_{r}$ (can be same).

For a finite group $G$, $r=0$ and $G\simeq \mathbb{Z}/p_{1}^{e_{1}}\times\cdots\times\mathbb{Z}/p_{r}^{e_{r}}$. Now you can classify those groups. For example, there are two order 4 abelian groups : $\mathbb{Z}/4$ and $\mathbb{Z}/2\times\mathbb{Z}/2$.