I recently proved this in one of NonCommAlgís Fun algebra problems. . Let be a nontrivial element in the intersection of all the nontrivial subgroups. If then has finite order. Otherwise and so for some and so and so again has finite order.
Suppose has order Now let be any element in the group If then has finite order. Otherwise is in the subgroup generated by and so for some integer It follows that so that has finite order.
NB: You can go further that the order of must be a prime and that every nonidentity element of has order a power