Let G be a group of order 42. 1. Determine the number of Sylow-7 subgroups of G

To determine the number of Sylow-7 subgroups of a group \( G \) of order 42, we can apply **Sylow's theorems**. Let's go through the steps:

### Step 1: Determine the Order of the Sylow-7 Subgroups
The order of a Sylow-7 subgroup is determined by the largest power of 7 that divides the order of the group \( |G| \). The order of \( G \) is 42, and we can factorize it as follows:

\[
|G| = 42 = 2 \times 3 \times 7
\]

So, the order of a Sylow-7 subgroup \( P_7 \) must be 7, as it is the largest power of 7 that divides 42.

### Step 2: Use Sylow's Theorem to Determine the Number of Sylow-7 Subgroups
Sylow's theorem tells us that the number of Sylow-7 subgroups, denoted by \( n_7 \), must satisfy two conditions:
1. \( n_7 \) divides the index \( |G| / |P_7| = 42 / 7 = 6 \).
2. \( n_7 \equiv 1 \mod 7 \).

### Step 3: List the Possible Divisors and Find \( n_7 \)
The divisors of 6 are 1, 2, 3, and 6. We need to find which of these values satisfy \( n_7 \equiv 1 \mod 7 \).

- \( n_7 = 1 \): \( 1 \equiv 1 \mod 7 \)
- \( n_7 = 2 \): \( 2 \not\equiv 1 \mod 7 \)
- \( n_7 = 3 \): \( 3 \not\equiv 1 \mod 7 \)
- \( n_7 = 6 \): \( 6 \not\equiv 1 \mod 7 \)

The only value that satisfies both conditions is \( n_7 = 1 \).

### Conclusion
There is exactly **one Sylow-7 subgroup** in \( G \).

Would you like any additional details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How many Sylow-2 subgroups can exist in a group of order 42?
2. If \( G \) is a group of order 42, what can you conclude about the structure of \( G \) if there is only one Sylow-7 subgroup?
3. How does the existence of a unique Sylow-7 subgroup affect the normality of this subgroup?
4. What can Sylow's theorems tell us about the number of Sylow-3 subgroups in \( G \)?
5. What is the significance of the condition \( n_p \equiv 1 \mod p \) in Sylow's theorems?

**Tip:** The uniqueness of a Sylow \( p \)-subgroup often implies that the subgroup is normal in the group, which can provide insights into the group's structure.

Explore the number of Sylow-7 subgroups in a group of order 42 using Sylow's theorems. Understand how group theory principles determine the structure of these subgroups and their significance within the group. Suitable for advanced undergraduate students and researchers.

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