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 ?
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!
2. Z_2 (+) Z_8
3. Z_4 (+) Z_4
4. Z_2 (+) Z_2 (+) Z_4
5. Z_2 (+) Z_2 (+) Z_2 (+) Z_2
3,4, and 5 satisfy the required condition, which is x+x+x+x=0 for each x in G.