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.