I am trying to prove that a group of order 12 must have an element of order 2.
I have a feeling this is a consequence of Lagrange's theorem.... although I am not sure how so.
The order of an element in a group of order 12 can be 1,2,3,4,6,or 12: the only element of order 1 is the unit, if has
order 12 then has order 2, and similarly if it has order 6 or 4 we get an element of order 2.
The only possibility left thus is that all the non-unit elements of G have order 3, but if we try to pair each element with its
unique inverse then we've a problem....take it from here.
Ps. Of course, if you already know Cauchy's Theorem or the Sylow theorems the problem is trivial, so I assume you don't.