- #1
rideabike
- 16
- 0
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[itex]\in[/itex]G : x is finite} with a,b [itex]\in[/itex]H.
Then a[itex]^{n}[/itex]=e and b[itex]^{m}[/itex]=e for some n,m.
And b[itex]^{-1}[/itex][itex]\in[/itex]H. (Can I just say this?)
Hence (ab[itex]^{-1}[/itex])[itex]^{mn}[/itex]=a[itex]^{mn}[/itex]b[itex]^{-mn}[/itex]=e[itex]^{m}[/itex]e[itex]^{n}[/itex]=e (Since G is abelian the powers can be distributed like that)
So ab[itex]^{-1}[/itex][itex]\in[/itex]H, and H≤G.
Last edited by a moderator: