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## Homework Statement

Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists.

## Homework Equations

/propositions/definitions[/B]Fundamental Theorem of Finite Abelian Groups

Lagrange's Theorem and its corollaries (not sure if helpful for this problem)

## The Attempt at a Solution

I used the Fundamental Theorem of Finite Abelian Groups to find the abelian groups. The prime factorization of 675 is

$$

\begin{split}

675 &= 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \\

& = 3^{2} \cdot 3 \cdot 5 \cdot 5 = 9 \cdot 3 \cdot 5 \cdot 5 \\

& = 3^{3} \cdot 5 \cdot 5 = 27 \cdot 5 \cdot 5 \\

&= 3 \cdot 3 \cdot 3 \cdot 5^{2} = 3 \cdot 3 \cdot 3 \cdot 25 \\

&= 3^{2} \cdot 3 \cdot 5^{2} = 9 \cdot 3 \cdot 25 \\

&= 3^{3} \cdot 5^{2} = 27 \cdot 25 .\\

\end{split}

$$

and the groups are

$$

\begin{split}

\mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5} \quad & \land \quad \mathbb{Z}_{9} \times \mathbb{Z}_{3} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5} \\

\mathbb{Z}_{27} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5} \quad & \land \quad \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{25} \\

\mathbb{Z}_{9} \times \mathbb{Z}_{3} \times \mathbb{Z}_{25} \quad & \land \quad \mathbb{Z}_{27} \times \mathbb{Z}_{25} .

\end{split}

$$

I am stuck on the second question "Find an element of order 45 in each one of the groups, if it exists.". I know that I have to find an ##a \in G## (where G is each of the above abelian groups) such that ##order(a) := \#(<a>) = \#(\{k \cdot a : k \in \mathbb{Z}\}) = 45##, where ##\#(\bullet)## is the cardinality of a set; or show that such an element a does not exist.

Hints are very much appreciated.

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