Group theory problem exceedinly difficult and no one can solve it. can you?

In summary, group theory is a challenging branch of mathematics that studies the properties of groups, which are mathematical structures composed of objects and operations. The difficulty of group theory lies in its abstract concepts and complex proofs. One example of a difficult problem in group theory is the Inverse Galois Problem, which has been unsolved for over 150 years and involves both abstract algebra and number theory. Despite attempts by renowned mathematicians, it is believed that this problem may be unsolvable. There are potential theories and conjectures that could lead to a solution, but their application to group theory is still being researched. While there may not be immediate practical applications to solving this problem, the pursuit of solving it has led to advancements in other areas of mathematics and
  • #1
betty2301
21
0

Homework Statement


~p.s. it should be H/N is abelian, not H being abelian.

Homework Equations


subgroup

The Attempt at a Solution


for a) i have some idea
for b) i have no idea.

help~
:)
 

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  • #2
Regarding (b), any subgroup of a solvable group must be solvable. (Why?) What interesting property do you know about a certain subgroup of [itex]S_n[/itex] for [itex]n \geq 5[/itex]?
 
  • #3
Regarding (a), what happens to the cycle structure of an element of [itex]S_n[/itex] if you conjugate it by another element?
 

1. What is group theory and why is it difficult?

Group theory is a branch of mathematics that studies the properties of groups, which are mathematical structures composed of objects and operations that follow specific rules. It is considered difficult because it involves abstract concepts and complex mathematical proofs.

2. Can you provide an example of a group theory problem that is exceedingly difficult?

One example of a difficult group theory problem is the Inverse Galois Problem, which asks whether every finite group can be realized as the Galois group of a finite extension of the rational numbers. This problem has been unsolved for over 150 years.

3. Why is it believed that no one can solve this problem?

The difficulty of the Inverse Galois Problem lies in the fact that it involves both abstract algebra and number theory, two fields of mathematics that are notoriously challenging. Additionally, the problem has been attempted by many renowned mathematicians without success, leading to the belief that it may be unsolvable.

4. Are there any theories or conjectures that could potentially solve this problem?

There are several theories and conjectures that could potentially lead to a solution for the Inverse Galois Problem, such as the Langlands program and the Riemann hypothesis. However, these are highly complex and unproven theories, and their application to group theory is still a subject of ongoing research.

5. Is there any practical application to solving this problem?

While the Inverse Galois Problem may not have immediate practical applications, the techniques and insights gained from attempting to solve it have led to advancements in other areas of mathematics and science. Additionally, the pursuit of solving such a difficult problem can push the boundaries of human knowledge and understanding, leading to new discoveries and innovations.

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