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.

Tonio

Ps. Of course, if you already know Cauchy's Theorem or the Sylow theorems the problem is trivial, so I assume you don't.