Let be a group, a prime number and . Prove that either is cyclic or where is the dihedral group of order .

**Solution**

It is clear for . So we will assume that is an odd prime. Choose with , . Let . Since every subgroup of index is normal, is a normal subgroup of and thus

for some integer . Note that, since is odd, and hence

Now

because . Thus and hence because . Therefore either or . So we will consider two cases:

- . In this case
and so . Thus is abelian and hence because is odd. So is cyclic in this case.

- . In this case
and so