## Group Theory problem

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

### Group Theory problem

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

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

thanks