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.