I'm self studying Abstract Algebra Theory and Application and would appreciate a proof check, as I never invoke the fact that

is an abelian group and the solution in the back of the book appears to be taking a different route than me, meaning my proof is likely wrong. If anyone could push me in the right direction I'd appreciate it.

Proposition: Let

be an abelian group. The elements of finite order in

form a subgroup

.

Proof. We know

is not empty as the order of the identity of

is one. We now must show that for all elements in

there exists an inverse element. Consider any element

of

and let

be its order. Thus,

, where

is the identity of

and now

. Hence, e=x

^{m}=x*x

^{m-1}=e, implying that x

^{m-1} = x

^{-1}.

The binary operation on T is associative as it is already associative on G.Quick notation clear up - * represents a binary operation.