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'll use Sylow's theorems.

### Step 1: Factorize the order of the group
The order of the group \( G \) is given by \( |G| = 42 \). First, we factorize 42:

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

### Step 2: Use Sylow's theorems
According to Sylow's theorems:

1. The number of Sylow-\( p \) subgroups (where \( p \) is a prime divisor of the order of the group) divides the order of the group and is congruent to 1 modulo \( p \).
2. Let \( n_p \) denote the number of Sylow-\( p \) subgroups.

For Sylow-7 subgroups:
- \( p = 7 \)
- \( |G| = 42 \), and the number of Sylow-7 subgroups \( n_7 \) divides \( \frac{42}{7} = 6 \).
- Additionally, \( n_7 \equiv 1 \pmod{7} \).

### Step 3: Possible values of \( n_7 \)
Since \( n_7 \) must divide 6 and satisfy \( n_7 \equiv 1 \pmod{7} \), we consider the divisors of 6:
- Divisors of 6: 1, 2, 3, 6

We now check each:
- \( n_7 = 1 \): \( 1 \equiv 1 \pmod{7} \) (satisfies the condition)
- \( n_7 = 2 \): \( 2 \not\equiv 1 \pmod{7} \) (does not satisfy the condition)
- \( n_7 = 3 \): \( 3 \not\equiv 1 \pmod{7} \) (does not satisfy the condition)
- \( n_7 = 6 \): \( 6 \not\equiv 1 \pmod{7} \) (does not satisfy the condition)

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

**Therefore, there is exactly 1 Sylow-7 subgroup in \( G \).**

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would you determine the number of Sylow-3 subgroups for a group of order 42?
2. What can be inferred about the structure of \( G \) if it has exactly one Sylow-7 subgroup?
3. How does the existence of a unique Sylow subgroup influence the structure of a group?
4. Can a group of order 42 be simple? Why or why not?
5. How can Sylow's theorems help determine whether a group of order 42 is abelian?

**Tip:** Remember, if there is exactly one Sylow-\( p \) subgroup in a group, it must be normal in the group.

Explore how to determine the number of Sylow-7 subgroups in a group of order 42 using Sylow's theorems. Understand the application of number theory and group theory concepts. Suitable for advanced undergraduate students studying group theory.

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