### **First Sylow Theorem:**

The **First Sylow Theorem** states that:

*If \( G \) is a finite group of order \( |G| = p^m \cdot n \), where \( p \) is a prime number and \( n \) is a positive integer not divisible by \( p \), then \( G \) contains a subgroup of order \( p^m \).*

This subgroup is called a **Sylow \( p \)-subgroup**.

### **Proof of the First Sylow Theorem:**

#### Step 1: Setup and action of \( G \) on subsets
Let \( G \) be a group with \( |G| = p^m \cdot n \), where \( p \) is prime and \( n \) is not divisible by \( p \). We aim to show that there exists a subgroup of \( G \) whose order is \( p^m \).

Consider the group \( G \) acting on the set \( X \) of all subsets of \( G \) that have exactly \( p^m \) elements. The number of elements in this set \( X \) is \( \binom{|G|}{p^m} = \binom{p^m \cdot n}{p^m} \). This number is not divisible by \( p \), because it contains \( n \) as a factor, and \( n \) is not divisible by \( p \).

#### Step 2: Orbit-Stabilizer Theorem and counting
Now, let \( G \) act on this set \( X \) by left multiplication. For any subset \( A \in X \), the size of the orbit of \( A \) under the action of \( G \) is given by the **Orbit-Stabilizer Theorem**:

\[
|G \cdot A| = \frac{|G|}{|Stab(A)|}
\]

where \( Stab(A) \) is the stabilizer of \( A \), i.e., the subgroup of elements in \( G \) that fix the subset \( A \).

#### Step 3: Analyze fixed points
Since the number of elements in \( X \) is not divisible by \( p \), there must be a fixed point under this action. This means there is a subset \( A \subseteq G \) of size \( p^m \) such that its stabilizer has order divisible by \( p^m \). Therefore, the stabilizer of this subset \( A \) is a subgroup of \( G \) of order \( p^m \).

#### Step 4: Conclusion
Thus, there exists a subgroup \( H \subseteq G \) such that \( |H| = p^m \). This subgroup is a **Sylow \( p \)-subgroup** of \( G \).

### **Conclusion:**
The First Sylow Theorem guarantees the existence of at least one Sylow \( p \)-subgroup of \( G \), i.e., a subgroup whose order is a power of the prime \( p \) (in this case, \( p^m \)).

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Do you want any further clarification or details on any part of the proof? 

Here are 5 related questions to deepen your understanding:

1. What are Sylow \( p \)-subgroups and why are they important in group theory?
2. How does the action of a group on a set help in proving the First Sylow Theorem?
3. What role does the Orbit-Stabilizer Theorem play in the proof of the First Sylow Theorem?
4. Can a group have more than one Sylow \( p \)-subgroup? If so, under what conditions?
5. How does the structure of the group \( G \) affect the number of Sylow \( p \)-subgroups?

**Tip:** Sylow's theorems are crucial for understanding the structure of finite groups, as they provide detailed information about subgroups related to prime factors of the group's order.

Explore the First Sylow Theorem in group theory, which guarantees the existence of a Sylow p-subgroup in a finite group. This detailed proof uses concepts like group actions and the Orbit-Stabilizer Theorem. Suitable for undergraduate students studying advanced algebra and group theory.

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