Do you know Lagrange's Theorem?
Prove that the dihedral group of order 6 does not have a subgroup of order 4.
Now I was really considering using the Caley Table and just listing all possible subgroups showing that one of order four does not exist but there has to be an easier way.
So i was thinking maybe a proof by contradiction.
Assume there is a subgroup of order four in the dihedral group of 6.
then find that an element x is not the identity and because it is a subgroup an identity element has to exist.
think fictional subgroup of order four that is not the identity. Say this is x. If x is a rotation what other elements must also be in the group?"