let and be the number of Sylow p-subgroups and q-subgroups respectively. we just need to show that either

m = 0 or n = 0. so suppose, on the contrary, that m > 0 and n > 0 and consider two possible cases:

Case 1: then we'll have and which is impossible.

Case 2: thus, since every two Sylow q-subgroups intersect in the identity element only, we'll have:

which is impossible.Q.E.D.

to show this, you'll need more than just this fact that for primes p, q, groups of order or or are not simple!2.if |G| < 100 and G is non-abelian and simple,then show that |G|=60.