The only subgroup of finite order of a an infinite group is the identity
My proof
Let G be a group with infinite order
let H be a subgroup and let and
which make H with an infinite order
am I rite ?
is there any counter example
More generally, take any field then if there exists non-zero with (i.e. a non-trivial root of unity) then the group of -roots of unity form a finite subgroup of . This clearly generalizes Dr. Revilla's example (so that, for example, the -roots of unity in ).
But, perhaps even more of a trivial counterexample. Take any infinite group and consider ....