1. ## Abelian Group Property

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:

(Q. 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 $G$?
Their choices are:

(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!

2. Originally Posted by Chris L T521
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:

(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!
Applying the FTFAG, the question is basically asking you how many abelian groups of order 16 are just direct products of $C_2$ and $C_4$, as these are the only cyclic groups with the given property.

Emm...I count 5...all $C_4$, or 3, 2, 1 or 0 $C_4$'s...

3. Originally Posted by Chris L T521
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:

(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.

4. Originally Posted by aliceinwonderland
Yes, it uses the fundamental theorem of finitely generated abelian groups. Belows are full classifications of order 16 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 satisfies the required condition, which is x+x+x+x=0 for each x in G.

5. Originally Posted by aliceinwonderland
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.
Ah, yes, that makes sense now. Thanks for the help!