# Congruence and p-sylow subgroups

• tom.young84
In summary, we can prove that [G:H]≡1 mod p by showing that N_H(P)=P, and then using Lagrange's theorem to conclude that |H| is divisible by |Px|, which has order p^k. This shows that [G:H]≡1 mod p.
tom.young84

## Homework Statement

Let p denote a prime number. Let G be a finite group. Let P denote a p-sylow subgroup of G. Finally, let H be any subgroup of G that contains N_G(P). Prove that [G]≡1 mod p.

## Homework Equations

Well I believe all we have to do here is conjugate stuff to show membership in various normalizers and then invoke Sylow III.

## The Attempt at a Solution

First off we know that for g $$\in$$ NG(H) such that gPg-1 and gHg-1. We also know that there exists a p-sylow subgroup Q=gPg-1. Since Q is a p-sylow subgroup of H, there exists an h in H such that hPh-1=Q. Now we see that ghP(gh)-1=Q. Subbing in something new, j=gh, jPj-1 is in NH(P) which is in H. This also shows, by moving stuff around, that NG(H) is in H. So, now we conclude H is in the Normalizer of H and the Normalizer is in H, so H=NG(H). The by sylow III, we get that ratio. Is this right, or close?

I was told there is a counting argument that uses group actions and this was more direct but a tad more complicated.

Last edited:

Your solution is on the right track, but there are a few errors and missing steps. Here is a possible solution:

First, we know that P is a p-sylow subgroup of G, so |P|=p^k for some k. Since P is a subgroup of G, it must also be a subgroup of H. Therefore, P is a p-sylow subgroup of H as well.

Next, we know that N_G(P) is the largest subgroup of G that contains P. Since H contains N_G(P), it follows that N_H(P) is also a subgroup of H. Therefore, N_H(P) is the largest subgroup of H that contains P, which means that N_H(P)=P.

Now, let x be any element of G. Since x is in G, it must also be in H, since H is a subgroup of G. This means that xPx^-1 is also a subgroup of H, and since P is a p-sylow subgroup of H, we must have xPx^-1=P. This shows that x is in N_H(P)=P.

Now, let g be any element of G. We know that gPg^-1 is a p-sylow subgroup of G, so it must also be a subgroup of H. This means that gPg^-1=P. Therefore, g is in N_H(P)=P.

Since g is in P and x is in P, it follows that gx is in P. This means that gx is in N_G(P), and since H contains N_G(P), it follows that gx is in H. Therefore, H contains the subgroup Px, which has order p^k.

Now, we can use Lagrange's theorem to conclude that |H| is divisible by |Px|=p^k, and therefore [G]≡1 mod p, as desired.

## 1. What is the definition of congruence in group theory?

Congruence in group theory refers to the property of two elements being equivalent with respect to a given subgroup. This means that the two elements behave in the same way when operating under the subgroup, and can be substituted for each other without changing the overall structure of the group.

## 2. How do p-Sylow subgroups relate to congruence?

p-Sylow subgroups are a type of subgroup that plays a significant role in determining the congruence of elements in a group. These subgroups have the property that they are the largest possible subgroups of a given order, and they also have a special relationship with the prime number p. The congruence of elements is often determined by their relationship with these p-Sylow subgroups.

## 3. Can congruence and p-Sylow subgroups be applied to any type of group?

Yes, congruence and p-Sylow subgroups can be applied to any type of group, including finite groups, infinite groups, abelian groups, and non-abelian groups. However, the specific properties and relationships between these concepts may vary depending on the type of group being studied.

## 4. How can congruence and p-Sylow subgroups be used in group theory applications?

Congruence and p-Sylow subgroups have many practical applications in group theory, including in the study of permutation groups, symmetry groups, and other algebraic structures. They can also be used to determine the structure and properties of various types of groups, and to classify groups into different categories based on their congruence and p-Sylow subgroups.

## 5. Are there any open research questions related to congruence and p-Sylow subgroups?

Yes, there are several ongoing research efforts related to congruence and p-Sylow subgroups in group theory. Some open questions include finding new methods for determining the congruence of elements, studying the relationship between different types of p-Sylow subgroups, and exploring the connections between congruence and other concepts in group theory such as cosets and normal subgroups.

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