Abelian Group Factored by Prime/Corollary

Corollary to Abelian Group Factored by Prime

A finite abelian group $G$ can be factored as follows:

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

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

This factorisation is unique up to ordering of the factors.

Proof

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

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 $\order K = \ds \prod_{i \mathop = 2}^k p_i^{n_i}$.

Therefore, it satisfies the induction hypothesis.

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

From:

Subgroup of Abelian Group is Normal

the definition of Sylow $p$-subgroup

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

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

$\blacksquare$