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Group Theory problem

Posted: Mon Jan 18, 2010 5:22 pm
by volodja
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?

Re: Group Theory problem

Posted: Mon Jan 18, 2010 6:41 pm
by ebaronov
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:
$$C_2\oplus C_2\oplus C_2\oplus C_2$$
$$C_4\oplus C_2\oplus C_2$$
$$C_4\oplus C_4$$

Re: Group Theory problem

Posted: Tue Jan 19, 2010 12:19 pm
by volodja
thanks :)