Prove that a non-abelian group of order 6 must have at least one element of period 3.
This is what I have so far.
Let in be a non-identity element. can have period 2, 3, or 6.
Period of cannot be 6 because that make cyclic, thus abelian.
If has period 3, we are done.
If has period 2, ... (stuck here)
Maybe looking at an element in and is not in , but not sure what to do with it.
Thanks in advance for any help.