(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This question is about sylow-p groups of Sp.

I've proved these parts of the question:

A. Each sylow p-sbgrp is from order p and there are (p-2)! p-sylow sbgrps of Sp.

B. (p-1)! = -1 (mod p ) [Wilson Theorem]

I need your help in these two :

C. 1) Let G be a group of order n*p^(k) where gcd(n,p)=1 and k>=1.

Prove that the normalizers of the p-sylow sbgrps of G are conjugated

C. 2) Make use of C.1), and find the order of the normalizer of a sylow-p-sbgrp of Sp.

2. Relevant equations

3. The attempt at a solution

About c1:

I proved it using the sylow-theorem that says that two sylow sbgrps are conjugated... I'm not sure that is the thing with the order.... Is it matter that the order is n*p^k? The argument is also true for an arbitrary group, no?

About c2:

I'm not that sure, but according to the theorem that says:

o(G)/o(N(P)) = Number of conjugates to P

We'll get that the order of the normalizer of N(P) (where P is a sylow-p-sbgrp) is p! (the order of Sp) divided by the number of conjugates to P, which are all the elements of the normalizers of the sylow-p-sbgrps...

Which means that it's p! / (p-2)!*p = (p-1)! / (p-2)! = p-1..

Verification and help are needed!!

TNX everyone!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Sylow p-groups

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**