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
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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!
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!