Hi

**pooma**.

I recently proved this in one of

**NonCommAlg**’s 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