If p is an odd prime, prove the following.
a) If G is a group of order , then G has a unique Sylow p-subgroup.
b) There are at least four groups of order which are pairwise nonisomorphic.
I know little about Sylow's subgroups.
You know that the number of Sylow -subgroups of divides and is equivalent to modulo . Now, tell me, how many numbers strictly less than any given are equivalent to modulo ?
Ok, once you take care of the obvious abelian ones, the important observation is that and so there exists non-trivial ____ products.b) There are at least four groups of order which are pairwise nonisomorphic.
it might be helpful to note that
and here i'm a bit confused. it seems to me, that this only gives 3 subgroups of order for the case p = 3.
so i believe that you also must investigate where you have a homomorphism into
as well.