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.

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- Dec 3rd 2011, 01:28 PMolgashukinaUnique Sylow p-Subgroups
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.

- Dec 3rd 2011, 03:53 PMDrexel28Re: Unique Sylow p-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 ?

Quote:

b) There are at least four groups of order which are pairwise nonisomorphic.

- Dec 3rd 2011, 05:13 PMDevenoRe: Unique Sylow p-Subgroups
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. - Dec 3rd 2011, 06:15 PMDrexel28Re: Unique Sylow p-Subgroups