We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.

Now let $ G $ be a finite group of order $ p^aq^b $ where $ p $ and $ q $ are prime numbers. Is this true that $ O^p (G) \ne G$ and also $ O^q (G)\ne G $? We note that by the solvability of $ G $ we know that one of them is not equal to $ G $.

Thanks for your help

obviouslynot. Sym(3) is a counterexample. Sym(3) has no nontrivial quotient which is a 3-group. $\endgroup$ – Johannes Hahn Mar 9 '14 at 22:43