What is the significance of p-sylow groups in finite groups?

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In summary, the homework statement is trying to prove that there is an element of order n in the quotient group N(P)/P. The attempt at a solution is to try to find this element by using m^n. However, the problem is that it is not clear how to find this element and it gets more complicated as the order of m gets smaller.
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TheForumLord
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Homework Statement


Let P be a p-sylow sbgrp of a finite group G.
N(P) will be the normalizer of P in G. The quotient group N(P)/P is cyclic from order n.

PROVE that there is an element a in N(P) from order n and that every element such as a represnts a generator of the quotient group N(P)/P


Homework Equations


The Attempt at a Solution



Welll... there is mP in N(P)/P such as (mP)^n = P -> m^n*P=P -> m^n = 1 ...
If m has order that is less the n, we'll get a contradiction to the fact that mP is from order n.
It's pretty obvious that every element of this kind is a generator of this group...But I really feel I'm missing something... It's a 20 points question and the answer will take me 2 lines...

Where is my mistake?

TNX in advance
 
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  • #2
You cannot deduce that [tex]m^n = 1[/tex] from [tex]m^n P = P[/tex]; that tells you only that [tex]m^n \in P[/tex]. However, there is a simple way to produce the element of order [tex]n[/tex] you need, using [tex]m^n[/tex].
 
  • #3
Hmmmm...Yep, you're right...
So we have m^n is in P... We need to produce an element k of order n that is in N(P)... Hmmmm we have m^n*P=P, and m^n is in P, m is in N(P). We know something more about this m? We know nothing about its order but we know that mP=Pm, which means m^n*P=P*m^n, and it's still gives us nothing...

Can you please give more detailed directions?
I'll appreciate any kind of further help...

TNX a lot
 
  • #4
Wait a sec! We know that m^n is in P and that P has order p^r for some r in N...So m^n must have an order that is a power of p, say p^k... so we know that m^(p^k) has order n in N(P)...
But how will we discover what k is? And why it's a generator of N(P)/P?


Will you help me please?
:(


TNX
 
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1. What is an abstract p-Sylow group?

An abstract p-Sylow group is a mathematical concept that refers to a subgroup of a finite group with a specific order, where p is a prime number. These subgroups are important because they provide information about the structure of the larger group.

2. How do you find the number of abstract p-Sylow groups in a finite group?

To find the number of abstract p-Sylow groups in a finite group, you can use the Sylow theorems, which state that the number of p-Sylow groups is congruent to 1 mod p, and that the number of p-Sylow groups divides the order of the group. By using these theorems, you can narrow down the possible number of p-Sylow groups and determine the exact number.

3. Can a group have more than one abstract p-Sylow group?

Yes, a group can have more than one abstract p-Sylow group, as long as the number of groups is congruent to 1 mod p. This is because the number of p-Sylow groups is determined by the order of the group and the prime number p, so there can be multiple groups with the same order and p.

4. What is the significance of abstract p-Sylow groups?

Abstract p-Sylow groups are significant because they provide important information about the structure of a larger group. They can also be used to prove other theorems and properties about groups, making them a valuable tool in mathematical research and problem-solving.

5. How are abstract p-Sylow groups related to other concepts in group theory?

Abstract p-Sylow groups are closely related to other concepts in group theory, such as normal subgroups, quotient groups, and the isomorphism theorems. They can also be used to prove other theorems and properties about groups, making them an important part of group theory as a whole.

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