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!
Yes, it uses the fundamental theorem of finitely generated abelian groups. Belows are the full classification of order 16 additive abelian groups up to isomorphism (see here).
1. Z_16
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.