Induction Method: My Question About Theorem in Finite Groups

In summary, the conversation discusses a theorem that states if a finite group G has a normal subgroup H and satisfies a certain statement, then G is solvable. In the proof, a counter example of minimal order is used and a proper normal subgroup N is introduced. It is mentioned that N satisfies the same statement as G and is solvable by induction in the order of G. However, there is a question about how the induction method was applied since H may not be a subgroup of N. An example of a similar theorem is also given, where H is a normal subgroup and all Sylow p-subgroups of G are conjugate in H. Again, a counter example of minimal order is used and a proper normal subgroup N is
  • #1
moont14263
40
0
My question is about the induction method. This was in a theorem that I read.
Let H be a normal subgroup of a finite group G. If G satisfies H-some statement then G is solvable.
In the poof I have this.
Let G be a counter example of minimal order. Let N be a proper normal subgroup of G. Since N satisfies the H-some statement then N is solvable by the induction in the order of G.


Here is my question. H may not be a subgroup of N so, how did he apply the induction method.
 
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  • #2
Here is an example of what I am talking about.
I made up this theorem.

Let H be a normal subgroup of a finite group G. If all Sylow p-subgroup P of G are conjugate in H then G is solvable.



Conjugate in H means the set {P^{h}:h [itex]\in[/itex] H} contain all Sylow p-subgroup of G where P is a Sylow p-subgroup of G.


Let G be a counter example of minimal order. Let N be a proper normal subgroup of G. I assume that all Sylow p-subgroup P of N are conjugate in H, "this is just an assumption ,it may not be true".then N is solvable by the induction in the order of G.


Here is my question. H may not be a subgroup of N so, how did he apply the induction method in his theorem which has the same situation ?.
 

1. What is the induction method?

The induction method is a mathematical proof technique used to prove that a statement is true for all natural numbers. It involves proving that the statement is true for the first few natural numbers, and then showing that if the statement is true for a particular number, it is also true for the next number. This process is repeated until the statement is proven to be true for all natural numbers.

2. How is the induction method used in finite groups?

The induction method can be used in finite groups to prove theorems about the structure and properties of these groups. It involves proving the theorem for the base case, usually the smallest possible group, and then using the group's structure and properties to show that if the theorem is true for a particular group, it is also true for the next group. This process is repeated until the theorem is proven to be true for all groups in the finite set.

3. What is the significance of using the induction method in finite groups?

The induction method is significant in finite groups because it allows for the efficient and systematic proof of theorems that hold true for all groups within a finite set. It also helps to reveal the underlying structure and properties of these groups, providing a deeper understanding of their behavior and potential applications.

4. Can the induction method be used in infinite groups?

No, the induction method can only be used in finite groups. This is because the method relies on the finite nature of the groups and their elements. In infinite groups, there is no smallest element or group to serve as the base case, making it impossible to apply the induction method.

5. What are some common mistakes made when using the induction method in finite groups?

One common mistake is assuming that the theorem is true for all groups without properly proving the base case. Another mistake is not considering all possible cases in the inductive step, leading to an incomplete or incorrect proof. It is important to carefully examine each step of the induction method to ensure a valid and comprehensive proof.

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