Notes on Cauchy's Theorem Presentation

Introduction

• The presentation focuses on the Cauchy's Theorem (also spelled as Ceelo's Theorem) and aims to provide intuitive understanding and proofs.
• Emphasis on obtaining examples of using these theorems.
• Requires background knowledge in group theory, specifically groups, orbits, stabilizers, and subgroups.

Cauchy's Theorem Overview

• Given a group G of size P^k * M, where P is a prime and P and M are coprime.
• The theorem states that G must have a subgroup of size P^k.

Importance of the Theorem

• Example with a group of size 3^2 * 5^2 * 7^2:
  • h1: subgroup of size 3^2
  • h2: subgroup of size 5^2
  • h3: subgroup of size 7^2
• Finding subgroups helps to understand the structure of G.
• If these groups are of prime square size, they are abelian.
• The goal is to piece together these subgroups to understand G.

Proof Strategy

• The proof utilizes a group action on a set.
• The set Ω consists of subsets of G of size P^k.
• Construct a group action by allowing G to act on Ω through left multiplication.

Group Action

• Action G on Ω:
  • G acts on subsets of size P^k by multiplication.
  • The image of this action maintains the size P^k.
  • Verify axioms of group action (not detailed in notes).

Analyzing Group Action

• Size of Ω:
  • Total elements = combinations of |G| choose |X| = (P^k * M) choose (P^k).
  • This is congruent to M mod P.
  • Since M is coprime with P, some orbits will have sizes not divisible by P.
• Selecting an orbit:
  • There exists an orbit of size not a multiple of P.

Orbit-Stabilizer Theorem

• From the Orbit-Stabilizer theorem:
  • Size of stabilizer * size of orbit = size of group G.
  • Since size of the orbit is not divisible by P, stabilizer must contain all P^k, implying that size of stabilizer divides P^k.

Size of Stabilizer

• The goal is to show size of stabilizer is exactly P^k.
• To do this, demonstrate that size of stabilizer is ≤ P^k.
• Randomly pick an element in X and analyze the map taking G to itself mapping it to G * X.
• This is a bijection.
  • Elements in the stabilizer keep the set X fixed, hence the outputs from stabilizer fall within X.
• The bijection establishes that the stabilizer size is ≤ size of X, where size of X = P^k.

Conclusion

• Combining results, the stabilizer is shown to be a subgroup of size P^k, fulfilling the theorem's requirement.
• Thus, G contains a subgroup of the desired size, confirming Cauchy's Theorem.