Generalization of Third Sylow Theorem

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This will give you a way to construct the desired subgroups and show that the number of them is congruent to 1 modulo p. In summary, to prove (a), you should use the proof of Sylow's theorem to construct subgroups of G containing H and then show that the number of these subgroups is congruent to 1 modulo p.
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glacier302
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Homework Statement



(a) Let G be a finite group that is divisible by by p^k, and suppose that H is a subgroup of G with order p^j, where j is less than or equal to k. Show that the number of subgroups of G of order p^k that contain H is congruent to 1 modulo p.

(b) Find an example of a finite group that has exactly p+1 Sylow p-subgroups.

Homework Equations



Theorem: Every p-group is contained in a Sylow p-subgroup.

Third Sylow Theorem: Let |G| = p^e*m where p does not divide m. Then the number of Sylow p-subgroups divides m and is congruent to 1 modulo p.

The Attempt at a Solution



I think that I should somehow be using the 3rd Sylow Theorem to prove (a). Also maybe the fact that since H is a p-group, it is contained in a Sylow p-subgroup, and any larger p-group containing H is also contained in a Sylow p-subgroup.

Any help would be much appreciated : )
 
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  • #2
Rather than trying to apply Sylow's theorem, you should try to apply (with appropriate modifications) its proof.
 

FAQ: Generalization of Third Sylow Theorem

1. What is the Third Sylow Theorem?

The Third Sylow Theorem is a mathematical theorem that states if a group G has a prime power order, then the number of Sylow p-subgroups of G is congruent to 1 modulo p.

2. How does the Third Sylow Theorem generalize?

The Third Sylow Theorem generalizes to a larger class of groups, including solvable groups and groups with certain types of composition series. It also applies to groups with non-prime power orders.

3. What is the significance of the Third Sylow Theorem?

The Third Sylow Theorem is an important tool in group theory, as it helps determine the structure and properties of finite groups. It also has applications in other areas of mathematics, such as algebraic geometry and number theory.

4. How does the Third Sylow Theorem relate to other Sylow theorems?

The Third Sylow Theorem is the most general of the three Sylow theorems, as it encompasses the First and Second Sylow Theorems as special cases. The First Sylow Theorem states that if a group G has a prime power order, then it has a subgroup of that order. The Second Sylow Theorem states that if G has a prime power order, then any two Sylow p-subgroups of G are conjugate.

5. Are there any open questions or further research related to the Third Sylow Theorem?

Yes, there are still open questions and ongoing research related to the Third Sylow Theorem. Some of these include generalizing the theorem to infinite groups and finding applications in other areas of mathematics. There is also ongoing work to understand the structure of groups that do not satisfy the conditions of the Third Sylow Theorem.

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