Theorem about p-groups, similar to 3rd Sylow Theorem

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In summary, the conversation discusses the properties of finite groups, specifically those divisible by p^k and with subgroups of order p^j. It is mentioned that the number of subgroups of G with order p^k containing H is congruent to 1 modulo p. An example of a finite group with exactly p+1 Sylow p-subgroups is also requested. The use of the 3rd Sylow Theorem and the 2nd Sylow Theorem are suggested as potential techniques for proving the statement in (a), along with the fact that p-groups have normal subgroups of all orders. Further assistance is requested.
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glacier302
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(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.

I think that I should be using the 3rd Sylow Theorem (the number of Sylow p-subgroups of G is congruent to 1 modulo p) to prove (a). Also maybe the fact that since H is a p-group, it is contained in Sylow p-subgroup by the 2nd Sylow Theorem, 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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The fact that p-groups have normal subgroups of all orders might prove useful, too.
 

FAQ: Theorem about p-groups, similar to 3rd Sylow Theorem

1. What is the Theorem about p-groups, similar to 3rd Sylow Theorem?

The Theorem about p-groups, similar to 3rd Sylow Theorem states that if G is a finite group and p is a prime number that divides the order of G, then G has a subgroup of order p^k for every k ≥ 0, where p^k is the highest power of p that divides the order of G.

2. What is the significance of the Theorem about p-groups, similar to 3rd Sylow Theorem?

The Theorem about p-groups, similar to 3rd Sylow Theorem is significant because it helps us understand the structure of finite groups and their subgroups. It also allows us to classify groups based on their order and the prime factors of their order.

3. How is the Theorem about p-groups, similar to 3rd Sylow Theorem proved?

The Theorem about p-groups, similar to 3rd Sylow Theorem is proved using a combination of group theory and combinatorics. It involves showing that the number of p-subgroups of G is congruent to 1 mod p, and using this result to construct a subgroup of order p^k for every k ≥ 0.

4. Can the Theorem about p-groups, similar to 3rd Sylow Theorem be extended to non-abelian groups?

Yes, the Theorem about p-groups, similar to 3rd Sylow Theorem can be extended to non-abelian groups. In fact, the original Sylow Theorems (1st, 2nd, and 3rd) were stated for non-abelian groups, but the 3rd Sylow Theorem is often referred to as the Theorem about p-groups because it is most commonly used for p-groups.

5. How is the Theorem about p-groups, similar to 3rd Sylow Theorem used in other areas of mathematics?

The Theorem about p-groups, similar to 3rd Sylow Theorem has applications in a wide range of mathematical areas, including algebraic geometry, number theory, and representation theory. It is also used in computer science, particularly in the study of error-correcting codes.

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