Thank you for your post. There are a few more questions concerning this and Sylow theorem that maybe you can help me with.
(i) What is the order of every Sylow 5-subgroup of ?
(ii) How many Sylow 5-subgroups does have?
For (i) I think the answer is 5 because
for (ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form and divides the order of , so I guess from this there is either 1 or 6 Sylow 5-subgroups in , so the answer is 6?? Can anyone verify any of this?
(iii) If is a Sylow p-subgroup of a finite group , and if is a Sylow p-subgroup of a finite group , prove that the direct product is a Sylow p-subgroup of the direct product .
For (iii) i attempted to prove it like follows: Take
Is this the right method? Thanks for any help anyone can provide.
of 5 cycles. for (iii) look at the orders: since P and Q are Sylow p-subgroups we have where then and
and clearly this completes the proof.
If you are allowed to assume that is simple, then the answer is immediately obvious. If there were just 1 Sylow 5-subgroup, this would be a proper nontrivial normal subgroup of since is simple, there must therefore be more than 1 Sylow 5-subgroup (indeed more than 1 Sylow subgroup of any order).