Thread: Sylow Subgroups

I'm not sure with the following:

(1) What is the order of every Sylow 5-subgroup of ?

(2) Let G and H be arbitrary finite groups. Prove that every sylow p-subgroup of has the form where is a sylow p-subgroup of G, and is a sylow p-subgroup of H.

(3) Find the number of sylow 5-subgroups of
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For (1) i think it's 25, as its's 5.5 ?

For (2) I think I use the Conjugacy sylow theorem somehow?

For (3) I think it might be 26 because is simple?

Thanks!

Originally Posted by Jason Bourne

I'm not sure with the following:

(1) What is the order of every Sylow 5-subgroup of ?

(2) Let G and H be arbitrary finite groups. Prove that every sylow p-subgroup of has the form where is a sylow p-subgroup of G, and is a sylow p-subgroup of H.

(3) Find the number of sylow 5-subgroups of
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For (1) i think it's 25, as its's 5.5 ?

For (2) I think I use the Conjugacy sylow theorem somehow?

For (3) I think it might be 26 because is simple?

Thanks!

Hi Jason Bourne.

(1) Your answer is correct.

(2) Determine the maximum power of dividing the order of

(3) No, it can’t be 26. The order of is The number of Sylow 5-subgroups must divide 26 does not divide 144.

for part (3) use part (2) and the fact that has 6 Sylow 5-subgroups. to prove that is a Sylow p-subgroup of if is a Sylow p-subgroup of and is a Sylow p-subgroup

do as TheAbstractionist suggested. but the converse is less trivial and it's the only fairly interesting part of your problem!