Let G be a group of order 45. 1. Determine the number of Sylow 3-subgroups of G.

To determine the number of Sylow 3-subgroups of a group \( G \) of order 45, we'll follow the steps using Sylow's theorems.

### Step 1: Factorize the order of the group
The order of the group \( G \) is given by \( |G| = 45 \). First, we factorize 45:

\[
45 = 3^2 \times 5
\]

### 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-3 subgroups:
- \( p = 3 \)
- \( |G| = 45 \), and the number of Sylow-3 subgroups \( n_3 \) divides \( \frac{45}{3^2} = 5 \).
- Additionally, \( n_3 \equiv 1 \pmod{3} \).

### Step 3: Possible values of \( n_3 \)
Since \( n_3 \) must divide 5 and satisfy \( n_3 \equiv 1 \pmod{3} \), we consider the divisors of 5:
- Divisors of 5: 1, 5

We now check each:
- \( n_3 = 1 \): \( 1 \equiv 1 \pmod{3} \) (satisfies the condition)
- \( n_3 = 5 \): \( 5 \equiv 2 \pmod{3} \) (does not satisfy the condition)

### Conclusion
The only value that satisfies both conditions is \( n_3 = 1 \).

**Therefore, there is exactly 1 Sylow-3 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-5 subgroups for a group of order 45?
2. What does it imply about \( G \) if it has a unique Sylow-3 subgroup?
3. How does the existence of a unique Sylow subgroup affect the normality of the subgroup in \( G \)?
4. What can you say about the structure of a group with a unique Sylow-3 subgroup?
5. Can a group of order 45 be simple if it has a unique Sylow-3 subgroup?

**Tip:** A unique Sylow-\( p \) subgroup in a group is always normal in that group, which has significant implications for the group's structure.

Explore how to determine the number of Sylow 3-subgroups in a group of order 45 using Sylow's theorems. Understand the implications for group structure and normality. Learn about divisors of 45 and their relevance to subgroup counts.

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