Thread: Sylow group question

1.Let G be a group with |G|=p^2*q,where p and q are distinct primes,show that G has a normal Sylow p-subgroups or a normal Sylow q-subgroups.

2.if |G| < 100 and G is non-abelian and simple,then show that |G|=60.

Please help to solve these two questions..

Originally Posted by seng

1.Let G be a group with |G|=p^2*q,where p and q are distinct primes,show that G has a normal Sylow p-subgroups or a normal Sylow q-subgroups.

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.

2.if |G| < 100 and G is non-abelian and simple,then show that |G|=60.

to show this, you'll need more than just this fact that for primes p, q, groups of order or or are not simple!

Actually i not really undestand how to do the second question,can you guide me to do?
I also will try my best to do...

Originally Posted by seng

Actually i not really undestand how to do the second question,can you guide me to do?
I also will try my best to do...

Just write out the numbers: .
And start out canceling the ones you know are not it.
Cross out the primes because you are looking for the non-abelian ones.
Now cross out the prime powers because -groups.
Now cross out the ones.
And then the ones.
And see if you can get rid of all those numbers.