Abelian Group Factored by Prime/Corollary

Corollary to Abelian Group Factored by Prime
Any finite abelian group $G$ can be factored as follows:

Let $\left|{G}\right| = \displaystyle \prod_{i \mathop = 1}^k p_i^{n_i}$ be the prime factorisation of the order of $G$.

Then we have $G = \displaystyle \prod_{i \mathop = 1}^k H_i$, where $H_i = \left\{{x \in G : x^{p_i^{n_i}} = e}\right\}$.

This factorisation is unique up to ordering of the factors.

Proof
Suppose $\left|{G}\right| = \displaystyle \prod_{i \mathop = 1}^k p_i^{n_i}$ is the prime factorisation of $\left|{G}\right|$.

We proceed by induction on $k$.

Basis for the induction
For $n = 1$, the statement is trivial.

Induction Hypothesis
Now we assume the theorem is true for abelian groups whose order has $k - 1$ distinct prime factors.

Induction Step
Apply Abelian Group Factored by Prime to $G$ and $p_1$.

By definition, $H = H_1$.

Also, the resulting $K$ has $\left|{K}\right| = \displaystyle \prod_{i \mathop = 2}^k p_i^{n_i}$.

Therefore, it satisfies the induction hypothesis.

It follows that $G = H_1 \times \displaystyle \displaystyle \prod_{i \mathop = 2}^k H_i$.

Furthermore, all the $H_i$ are normal Sylow $p$-subgroups.

From Normal Sylow P-Subgroup is Unique, the factorisation is unique up to ordering of the factors.