finitely generated Abelian groups

• Apr 17th 2009, 02:30 PM
georgel
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.
• Apr 17th 2009, 05:23 PM
aliceinwonderland
Quote:

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.
• Apr 18th 2009, 12:45 AM
georgel
Quote:

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..
• Apr 18th 2009, 01:07 AM
aliceinwonderland
Quote:

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}\$.
• Apr 19th 2009, 02:34 AM
georgel
Quote:

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!