1. finitely generated Abelian groups

Identify all finitely generated Abelian groups of order 500 that have exactly 3 elements of order 2 and identify those elements.

I think that finitely generated Abelian groups of order 500 are:

$\displaystyle \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o plus\mathbb{Z}_5\$

$\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o plus\mathbb{Z}_5\$

$\displaystyle \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25 }\$

$\displaystyle \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_{125}$

$\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12 5}$

$\displaystyle \mathbb{Z}_4\oplus \mathbb{Z}_{125}$

Is this correct?
And if it is, which ones have exactly 3 elements of order 2?

Any help or hint is appreciated.

2. Originally Posted by georgel
Identify all finitely generated Abelian groups of order 500 that have exactly 3 elements of order 2 and identify those elements.

I think that finitely generated Abelian groups of order 500 are:

$\displaystyle \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o plus\mathbb{Z}_5\$

$\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_5\o plus\mathbb{Z}_5\$

$\displaystyle \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25 }\$

$\displaystyle \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus\mathbb{Z}_{125}$

$\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12 5}$ (wrong !)

$\displaystyle \mathbb{Z}_4\oplus \mathbb{Z}_{125}$

Is this correct?
And if it is, which ones have exactly 3 elements of order 2?

Any help or hint is appreciated.
The below one seems like a typo.
$\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12 5}$ should be changed to $\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25 }$.
Other than that, your classification looks OK to me.

3. Originally Posted by aliceinwonderland
The below one seems like a typo.
$\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{12 5}$ should be changed to $\displaystyle \mathbb{Z}_4\oplus\mathbb{Z}_5\oplus\mathbb{Z}_{25 }$.
Other than that, your classification looks OK to me.
Yes, it's a typo, thank you.

Do you know how to determine which ones have 3 elements of order two? I know how to check whether element is of order two, but the idea is hardly to write down all elements and check..

4. Originally Posted by georgel
Yes, it's a typo, thank you.

Do you know how to determine which ones have 3 elements of order two? I know how to check whether element is of order two, but the idea is hardly to write down all elements and check..
In $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{250}$, there are three elements of order 2, which are

(1,0), (0, 125), (1, 125).

Since $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{250}$ is isomorphic to $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}$, there are three elements of order 2 in $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}$.

5. Originally Posted by aliceinwonderland
In $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{250}$, there are three elements of order 2, which are

(1,0), (0, 125), (1, 125).

Since $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{250}$ is isomorphic to $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}$, there are three elements of order 2 in $\displaystyle \mathbb{Z}_2 \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{125}$.
Thank you again, I see that they are elements of order 2, I just fail to see how to get them myself. I'll think about it a bit longer.

Thank you!