I'm currently reviewing for the GRE Math Subject Test and am just about done with the practice exam supplied by ETS. I'm having trouble understanding the solution to one question:

Their choices are:(Q. 49)

Up to isomorphism, how many additive abelian groups of order 16 have the property that for each in ?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 5

And they say that (D) is the correct answer. I'm trying to understand why. Do they use the fundamental theorem of finite abelian groups somehow?

I would appreciate if anyone could shed some light on this!