Intersection of Sylow p-subgroups of order 42 And subgroup of order 6

Given that 42 is the product of distinct prime factors \( 2 \times 3 \times 7 \), it's important to correctly identify the nature of the Sylow subgroups and their intersections in any group \( G \).

### Sylow \( p \)-subgroups and Their Intersections

For a finite group \( G \):
1. The Sylow 2-subgroup \( P \) has order \( 2^1 = 2 \).
2. The Sylow 3-subgroup \( Q \) has order \( 3^1 = 3 \).
3. The Sylow 7-subgroup \( R \) has order \( 7^1 = 7 \).

These Sylow subgroups \( P, Q, R \) are typically maximal \( p \)-subgroups corresponding to each prime factor \( p \).

### Intersecting Sylow Subgroups

The intersection of Sylow \( p \)-subgroups for distinct primes is generally trivial, meaning they only share the identity element. This stems from the fact that Sylow subgroups for different primes do not contain common non-identity elements.

### Subgroup of Order 6

A subgroup of order 6 is non-trivial and can be constructed as follows:
- A subgroup \( H \) of order 6 has order \( 2 \times 3 \).

In a group \( G \) of order 42, such a subgroup \( H \) can be formed as a product of Sylow subgroups \( P \) and \( Q \), where \( P \) is of order 2 and \( Q \) is of order 3. This is possible if the group has a specific structure that allows such a combination, e.g., a semi-direct product.

### Intersection of Subgroups

1. **Intersection of Sylow \( p \)-subgroups:**
   - Intersection of \( P \), \( Q \), and \( R \) (of orders 2, 3, and 7 respectively) in \( G \) is trivial if \( G \) is simple and doesn't have normal subgroups. That is,
   \[ P \cap Q \cap R = \{e\} \]

2. **Intersection with Subgroup of Order 6:**
   - Let \( H \) be the subgroup of order 6.
   - The elements of \( H \) are combinations of elements from \( P \) and \( Q \).
   - The intersection of \( H \) with any Sylow \( p \)-subgroup would typically be:
     - \( H \cap P \) is a Sylow 2-subgroup of \( H \) and thus of order 2.
     - \( H \cap Q \) is a Sylow 3-subgroup of \( H \) and thus of order 3.
   - The elements of \( H \) are combinations of elements from \( P \) and \( Q \), so:
     - \( H \cap P = P \), if \( P \subseteq H \).
     - \( H \cap Q = Q \), if \( Q \subseteq H \).

In summary:
- The intersection of the Sylow \( p \)-subgroups \( P, Q, \) and \( R \) is trivial.
- The intersection of \( H \) (subgroup of order 6) with \( P \) or \( Q \) will be non-trivial, containing elements that combine their orders.

### Would you like more details or have any specific questions on this topic?

### Related Questions
1. What is the intersection of all Sylow subgroups in a simple group?
2. How do you construct a subgroup of a specific order in a given finite group?
3. Can you explain the Sylow theorems in detail with examples?
4. What is the significance of non-trivial intersections of subgroups?
5. How do the normalizers of Sylow subgroups relate to their intersections?
6. What is the role of the center of a group in the context of Sylow subgroups?
7. Can two Sylow subgroups of different orders intersect non-trivially?
8. How does the structure of a group influence the behavior of its Sylow subgroups?

### Tip
When dealing with intersections of subgroups, always consider the order of the subgroups involved and their relative prime factors. This will help in understanding whether the intersection is trivial or non-trivial.

Explore the intersection of Sylow p-subgroups of order 42 and subgroups of order 6 in finite groups. Understand the implications of Sylow theorems in group theory and how subgroups of different orders intersect within specific group structures.

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