Group Theory problem

Forum for the GRE subject test in mathematics.
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volodja
Posts: 5
Joined: Mon Jan 18, 2010 5:18 pm

Group Theory problem

Post by volodja » Mon Jan 18, 2010 5:22 pm

Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G?

ebaronov
Posts: 5
Joined: Wed Jan 06, 2010 10:49 am

Re: Group Theory problem

Post by ebaronov » Mon Jan 18, 2010 6:41 pm

Every finite abelian group can be expressed as as direct sum of cyclic groups. Since x+x+x+x=0 for each x, the order of each of these cyclic groups is 2 or 4. We have 3 possibilities:
$$C_2\oplus C_2\oplus C_2\oplus C_2$$
$$C_4\oplus C_2\oplus C_2$$
$$C_4\oplus C_4$$

volodja
Posts: 5
Joined: Mon Jan 18, 2010 5:18 pm

Re: Group Theory problem

Post by volodja » Tue Jan 19, 2010 12:19 pm

thanks :)



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