Let 𝐺 be a group of order 52 = 4 ⋅ 13.
1. How many subgroups of order 13 does 𝐺 have?

Given that \( G \) is a group of order 52, we can determine the number of subgroups of order 13 using the Sylow theorems.

### Step 1: Apply the First Sylow Theorem
The First Sylow Theorem states that for a prime \( p \), there exists at least one Sylow \( p \)-subgroup of \( G \), where \( p^n \) divides the order of the group \( |G| \). 

Here, \( |G| = 52 = 4 \times 13 \), so the primes to consider are \( p = 2 \) and \( p = 13 \).

### Step 2: Focus on Sylow \( 13 \)-Subgroups
We are interested in the subgroups of order 13. According to the Sylow theorems, the number of Sylow \( p \)-subgroups, denoted by \( n_p \), must satisfy:
\[ n_{13} \equiv 1 \pmod{13} \]
\[ n_{13} \mid \frac{|G|}{13} = \frac{52}{13} = 4 \]

So, \( n_{13} \) must be a divisor of 4 and satisfy \( n_{13} \equiv 1 \pmod{13} \).

### Step 3: Determine the Possible Values of \( n_{13} \)
The divisors of 4 are 1, 2, and 4. Now, we check which of these satisfy \( n_{13} \equiv 1 \pmod{13} \):

- \( 1 \equiv 1 \pmod{13} \) — This is true.
- \( 2 \equiv 2 \pmod{13} \) — This is not true.
- \( 4 \equiv 4 \pmod{13} \) — This is not true.

Thus, \( n_{13} = 1 \).

### Conclusion
There is exactly **one** Sylow 13-subgroup of \( G \). 

#### Do you want details or have any questions?
Here are some related questions to expand your understanding:
1. How can you use Sylow theorems to find the number of subgroups of a different prime order?
2. What is the significance of the Sylow theorems in group theory?
3. Can you determine how many Sylow 2-subgroups exist in \( G \)?
4. What is the order of the automorphism group of a cyclic group of order 13?
5. If a group has exactly one Sylow \( p \)-subgroup, what can you conclude about this subgroup?
6. How do Sylow subgroups relate to the structure of a group?
7. Could the group \( G \) be simple? Why or why not?
8. What is the role of the center of a group in determining its subgroups?

**Tip:** When dealing with Sylow theorems, always check the divisibility condition and the modularity condition carefully, as they guide you to the correct number of subgroups.

Explore how to determine the number of subgroups of order 13 in a group of order 52 using Sylow theorems. Detailed step-by-step solution and explanations included. Suitable for advanced undergraduate students and enthusiasts of group theory.

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