gre 0809 #49

[Constantin]

Help me please solve the following task

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

Thanks in advance!

[lunarmono]

First of all, 16 factors as 2^4, so ALL the abelian groups of order 16 are:

Z2 x Z2 x Z2 x Z2
Z4 x Z2 x Z2
Z4 x Z4
Z8 x Z2
Z16

The last two will have elements of order greater than 4, namely (1,1) in Z8 x Z2 which will be of order 8 and 1 in Z16 which is of order 16. In all the other groups, every nonzero element is of order 4 or 2 and so will satisfy x+x+x+x=0.

[Constantin]

Thanks a lot!