Proving Finite Order Elements Form a Subgroup of an Abelian Group

[rideabike]

Homework Statement

Prove the collection of all finite order elements in an abelian group, G, is a subgroup of G.

The Attempt at a Solution

Let H={x$\in$G : x is finite} with a,b $\in$H.
Then a$^{n}$=e and b$^{m}$=e for some n,m.
And b$^{-1}$$\in$H. (Can I just say this?)
Hence (ab$^{-1}$)$^{mn}$=a$^{mn}$b$^{-mn}$=e$^{m}$e$^{n}$=e (Since G is abelian the powers can be distributed like that)

So ab$^{-1}$$\in$H, and H≤G.

 

[jbunniii]

Homework Statement

Prove the collection of all finite order elements in an abelian group, G, is a subgroup of G.

The Attempt at a Solution

Let H={x$\in$G : x is finite} with a,b $\in$H.
Then a$^{n}$=e and b$^{m}$=e for some n,m.
And b$^{-1}$$\in$H. (Can I just say this?)

If you're going to say it, you should justify it. But your proof below doesn't use this fact. (Indeed, it proves it!)

Hence (ab$^{-1}$)$^{mn}$=a$^{mn}$b$^{-mn}$=e$^{m}$e$^{n}$=e (Since G is abelian the powers can be distributed like that)

So ab$^{-1}$$\in$H, and H≤G.

This part is fine. Note that as a special case, this shows that $b^{-1} \in H$. (Take $a = e$.) You didn't need to stipulate $b^{-1}\in H$ in the previously quoted section.

 

[Evo]

Delete thread, figured it out

We don't delete threads once they have a response.

Thank you jbunniii!