The question is as stated in the title. I used LaGrange's theorem to show that |H|||G| and |K|||G| so 12||G| and 5||G|, and that 12 and 5 are relatively prime. I'm not sure if this has gotten me closer to solving the problem, as I am stuck. Any help is greatly appreciated.
NOTE: This problem provides does not specifically define e, so I'm attempting to prove this assuming e is the identity of G.