let G be a group of order 42.

Determine the number of Sylow-7 subgroups of G.


Given that \( G \) is a group of order 42, we need to determine the number of Sylow-7 subgroups.

First, factorize the order of the group:
\[ |G| = 42 = 2 \times 3 \times 7 \]

According to the Sylow theorems, the number of Sylow-\( p \) subgroups, denoted \( n_p \), must satisfy two conditions:
1. \( n_p \equiv 1 \pmod{p} \)
2. \( n_p \) divides \( |G| / p \)

For the Sylow-7 subgroups:
- \( p = 7 \)
- \( |G| / 7 = 6 \)

So, \( n_7 \) must divide 6 and \( n_7 \equiv 1 \pmod{7} \).

The divisors of 6 are 1, 2, 3, and 6. We need to check which of these satisfy \( n_7 \equiv 1 \pmod{7} \):

- \( n_7 = 1 \): \( 1 \equiv 1 \pmod{7} \) ✔️
- \( n_7 = 2 \): \( 2 \equiv 2 \pmod{7} \) ❌
- \( n_7 = 3 \): \( 3 \equiv 3 \pmod{7} \) ❌
- \( n_7 = 6 \): \( 6 \equiv 6 \pmod{7} \) ❌

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

Therefore, the number of Sylow-7 subgroups in \( G \) is:
\[ n_7 = 1 \]

This means there is exactly one Sylow-7 subgroup in \( G \).

Do you have any questions or want more details?

Here are some related questions to further your understanding:

1. What is the significance of the Sylow theorems in group theory?
2. How do we determine the number of Sylow-\( p \) subgroups for other prime factors?
3. What does it mean for a subgroup to be normal in a group?
4. Can there be more than one Sylow-2 subgroup in \( G \)? Why or why not?
5. How does the existence of a unique Sylow-7 subgroup affect the structure of \( G \)?
6. What is the order of the Sylow-3 subgroups in \( G \)?
7. How are the Sylow subgroups used to analyze the structure of finite groups?
8. Can the number of Sylow-7 subgroups ever be zero? Why or why not?

**Tip:** Understanding the interplay between Sylow subgroups and group order helps in analyzing the internal structure and classification of finite groups.

Explore how to determine the number of Sylow-7 subgroups in a group of order 42 using Sylow theorems and modular arithmetic. Understand the conditions and calculations involved in finding the unique Sylow-7 subgroup.

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