Let be a group. Prove that if then is not simple.
Note that . Note that the divisibility and congruence criteria of Sylow’s Theorem force the following.
Note in particular that for each prime dividing 6545, the Sylow -subgroups of intersect trivially. If no Sylow subgroup is normal, then has at least elements, a contradiction. Thus some Sylow subgroup of is normal, and hence is not simple.