# Thread: Number of subgroups of an abelian group

1. ## Number of subgroups of an abelian group

How many subgroups of order $p^{2}$ does the abelian group $Z_{p^{3}} \oplus Z_{p^{2}}$ have?

2. ## Some things to consider

A subgroup of an abelian group inherits commutativity, and we know the order of the subgroup is $p^2$, so we can apply the fundamental theorem of finitely generated groups to see that the only groups of order $p^2$ are $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$. All that it amounts to is counting elements of order $p$ and $p^2$ in the two groups, and then I think you can take it from there to count how many subgroups of this order you could make out of these elements.

Hint: Theorem If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is $\phi(d)$ where $\phi$ is Euler's totient function. Euler's totient function - Wikipedia, the free encyclopedia it is the number of positive integers less than and relatively prime to n.