Question 1
What is the significance of finding subgroups of prime square size in the context of Cauchy's Theorem?
Question 2
What does it imply about a group G if it has a subgroup of prime square size?
Question 3
What is the primary set Ω used for in the proof of Cauchy's Theorem?
Question 4
What is the statement of Cauchy's Theorem in group theory?
Question 5
How is the group action defined in Cauchy's Theorem proof?
Question 6
In the provided example within the student's notes, what are the subgroup sizes associated with each prime factor?
Question 7
Why do we consider subsets of G of size P^k in Cauchy's Theorem?
Question 8
What role does the coprimality of P and M play in the proof of Cauchy's Theorem?
Question 9
What key theorem is utilized to show that a non-divisible by P orbit exists in Cauchy's proof?
Question 10
In the context of Cauchy's Theorem, why must some orbits under group action have sizes not divisible by P?
Question 11
How does the existence of a subgroup of size P^k affect structural analysis of G?
Question 12
How is the bijection related to the size of the stabilizer?
Question 13
How does the Orbit-Stabilizer Theorem help in proving Cauchy's Theorem?
Question 14
Why is it necessary for the image of group action on Ω to maintain the size P^k?
Question 15
What must be proven about the stabilizer to complete the proof of Cauchy's Theorem?