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.