An Interesting Question_GR0568 #49

Forum for the GRE subject test in mathematics.
weilan8
Posts: 8
Joined: Sun Aug 16, 2009 12:19 pm

An Interesting Question_GR0568 #49

Postby weilan8 » Fri Oct 02, 2009 5:28 pm

Hi everyone, thank you for interest to solve this problem:

49. 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? (A)0 (B)1 (C)2 (D)3 (E)5

order(<x>)=4. So according to the theories about elementary divisor or the invariant factors of G, G could be: (here "+" stands for any additive operation not just the ordinary one)
Z2+Z2+Z2+Z2
Z2+Z2+Z4
Z2+Z8
Z4+Z4
Z16
It looks like there are 5. However, the answer is (D). I think the condition that "x+x+x+x=0 for each x in G" must have some constrain. But, I don't know how to use this constrain to choose the correct groups.

prong
Posts: 24
Joined: Thu Sep 24, 2009 12:17 am

Re: An Interesting Question_GR0568 #49

Postby prong » Fri Oct 02, 2009 8:47 pm

Are you sure Z16 works? What about Z2 + Z8?

In fact 1 + 1 + 1 + 1 = 4 != 0 in Z16, and (0,1) + (0,1) + (0,1) + (0,1) = (0,4) != 0 in Z2 + Z8.

weilan8
Posts: 8
Joined: Sun Aug 16, 2009 12:19 pm

Re: An Interesting Question_GR0568 #49

Postby weilan8 » Sat Oct 03, 2009 4:45 pm

Thanks for your hint!

After second thought, x+x+x+x=0 not only implies order(x)=4, but also indicates x=0 or order(x)=2. Therefore, the candidate group will be:
Z2+Z2+Z2+Z2, which contains (0,0,0,0) or other elements with the order of 2
Z2+Z2+Z4, which contains (0,0,0) or other element with the order of 2 or 4
Z4+Z4, which contains (0,0) or other elements with the order of 2 or 4




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