# Verifying Sylow-p Groups of Sp: A Homework Exercise

• TheForumLord
In summary, the conversation discussed the proof of two parts of a question about Sylow p-groups of Sp. The first part involved proving that the normalizers of the p-sylow subgroups of a group of order n*p^k are conjugate, and the second part involved finding the order of the normalizer of a Sylow p-subgroup of Sp using a theorem about conjugates. The summary also confirms the correctness of the proofs and reasoning provided by the individual.
TheForumLord

## Homework Statement

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.

## The Attempt at a Solution

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?

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!

I can confirm that your proof for C.1) is correct. The Sylow theorem states that any two Sylow p-subgroups are conjugate, which means that their normalizers must also be conjugate. This is true for any group, not just Sp.

For C.2), your reasoning is also correct. The order of the normalizer of a Sylow p-subgroup will be equal to p! divided by the number of conjugates to P, which is equal to (p-1)! divided by the number of Sylow p-subgroups, which we already know to be (p-2)! according to A. Therefore, the order of the normalizer of a Sylow p-subgroup of Sp is (p-1)! / (p-2)! = p-1.

Great job on your proof and reasoning! Keep up the good work.

## 1. What is a Sylow-p group?

A Sylow-p group is a group of order p^n, where p is a prime number and n is a positive integer. It is a subgroup of a larger finite group and has the property that it is the largest subgroup of the given order that can be formed within the larger group.

## 2. What is the significance of verifying Sylow-p groups of Sp?

The group Sp, or the symplectic group, is an important mathematical object that has applications in physics, engineering, and other areas. By verifying the existence and properties of Sylow-p groups within Sp, we gain a better understanding of the structure and behavior of this group, which can lead to further insights and applications.

## 3. How do you verify a Sylow-p group of Sp?

To verify a Sylow-p group of Sp, we need to show that it satisfies three conditions: it has the correct order, it is a subgroup of Sp, and it is a maximal subgroup of Sp with the given order. This can be done by using techniques such as group theory, combinatorics, and linear algebra.

## 4. What are some common techniques used in verifying Sylow-p groups of Sp?

Some common techniques include counting elements, using Sylow's theorems, and applying properties of finite fields. These techniques involve using the properties of groups, prime factorization, and linear algebra to verify the existence and properties of Sylow-p groups within Sp.

## 5. Why is verifying Sylow-p groups of Sp important for research and applications?

Verifying Sylow-p groups of Sp is important for further research and applications because it helps us understand the structure and properties of Sp, which can have implications in other areas of mathematics and science. Additionally, it can also lead to the development of new techniques and algorithms for solving complex problems involving finite groups.

Replies
6
Views
1K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
5
Views
2K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
9
Views
2K
Replies
1
Views
2K
Replies
1
Views
3K
Replies
5
Views
3K