Hey I am having trouble with these problems:





For the first question I already got the first part and just need help with the second part where its for the general case.

Thank you!

How did you get the first part ?

The general case should fall quickly if you did it with Sylow theorems :

Consider Np the number of p-Sylow subgroups in G, where |G| = n = p.k
According to Sylow theorems, Np divides k, and Np = 1 modulo p.
Hence, obviously, Np = 1.
It means that there exists only 1 p-Sylow, that must be normal, because it coincides with all its conjugates (because they are also p-Sylows).

I found that 88 = 2^3 * 11 and it ended up that there was only one Sylow 11-subgroup. Thank you!