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Let $G$ be an abelian group of order $m > 1$. Let the prime power factorization of $m$ be given by $m = p_{1}^{\alpha_{1}} \cdots p_{k}^{\alpha_{k}}$. Then:

(a) $G \cong H_{1} \times H_{2} \times \cdots \times H_{k}$ where $|H_{i}| = p_{i}^{\alpha_{i}}$ for $1 \leq i \leq k$ and $i \in \mathbb{N}$.

(b) for every $H \in \{H_{1}, H_{2}, ..., H_{k}\}$ with $|H| = p^{\alpha}$, we have that $H \cong C_{p^{\beta_{1}}} \times C_{p^{\beta_{2}}} \times \cdots \times C_{p^{\beta_{s}}}$ where each $C_{p^{\beta{j}}}$ are cyclic groups of order $p^{\beta_{j}}$, $j \in \mathbb{N}$, $1 \leq j \leq s$, $\beta_{1} \geq \beta_{2} \geq \cdots \geq \beta_{s} \geq 1$, $\beta_{1} + \beta_{2} + \cdots + \beta_{s} = \alpha$, and $s, \beta_{1}, ..., \beta_{s}$ depend on $i$.

(c) the decomposition in $(a)$ and $(b)$ is unique.

How many abelian groups are there of order $2700$?









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