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# Finite Abelian Groups and the Fundamental Theorem of Arithmetic

NUMTH-KCHSMH

Let $k \in \mathbb{N}$. Then, $\mathbb{Z}/k\mathbb{Z} = \{ [a]_k | 0 \leq a < k, a \in \mathbb{Z}\}$, where $[a]_k = \{a + bk | b \in \mathbb{Z}\}$. If $n \in \mathbb{N} \backslash \{1\}$ and $n = p_{1}^{\alpha_1} \cdots p_{s}^{\alpha_{s}}$, where $p_1, ..., p_{s}$ are distinct prime numbers and $\alpha_1, ... \alpha_{s} \in \mathbb{N} \cup \{0\}$, for some $s \in \mathbb{N}$, then $\mathbb{Z} / n\mathbb{Z} \cong \mathbb{Z} /p_{1}^{\alpha_1}\mathbb{Z} \times \cdots \times \mathbb{Z} / p_{s}^{\alpha_{s}}\mathbb{Z}$. What is $\mathbb{Z} / 36\mathbb{Z}$ isomorphic to?

A

$\mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 9\mathbb{Z}$

B

$\mathbb{Z} / 36\mathbb{Z}$

C

$\mathbb{Z} / 4\mathbb{Z} \times \mathbb{Z} / 9\mathbb{Z}$

D

$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z} / 3\mathbb{Z} \times \mathbb{Z} / 3\mathbb{Z}$