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

To determine the number of subgroups of order 13 in a group \( G \) of order 52, we can apply Sylow's theorems. These theorems provide insights into the number and structure of subgroups of particular orders in a finite group.

### Step-by-Step Solution

1. **Identify the Orders of Possible Sylow Subgroups:**

   Since the order of the group \( |G| = 52 \), we can find the Sylow subgroups for the prime factors of 52.

   \[
   52 = 2^2 \times 13
   \]

   We will consider Sylow \( p \)-subgroups for each prime divisor \( p \) of 52: 2 and 13.

2. **Consider Sylow 13-subgroups:**

   By Sylow's theorems, the number of Sylow 13-subgroups, denoted \( n_{13} \), must satisfy:

   \[
   n_{13} \equiv 1 \pmod{13}
   \]
   \[
   n_{13} \mid \frac{52}{13} = 4
   \]

   Therefore, the divisors of 4 are 1 and 4. Checking these divisors:

   - If \( n_{13} = 1 \), then \( 1 \equiv 1 \pmod{13} \), which is true.
   - If \( n_{13} = 4 \), then \( 4 \equiv 1 \pmod{13} \), which is false.

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

3. **Conclusion:**

   There is exactly one Sylow 13-subgroup in \( G \).

The group \( G \) has exactly **one subgroup of order 13**.

### Additional Questions for Exploration

1. What are Sylow's theorems and how are they applied in group theory?
2. Can a group of order 52 be simple? Why or why not?
3. What is the significance of Sylow subgroups in understanding the structure of a group?
4. How do the Sylow subgroups relate to the overall composition of the group \( G \)?
5. If \( G \) has a unique Sylow 13-subgroup, what can we say about its normality in \( G \)?
6. How would the presence of more than one Sylow subgroup of a certain order affect the group's structure?
7. What are some real-world applications where understanding the subgroup structure of a group is essential?
8. How do you determine the number of subgroups of a particular order if the group is abelian?

**Tip:** When analyzing the structure of finite groups, understanding the order and number of Sylow subgroups can provide valuable insights into the group's composition and symmetries.

Explore how to determine the number of subgroups of order 13 in a group of order 52 using Sylow's theorems. Detailed step-by-step solution provided, covering the application of group theory concepts and Sylow subgroups.

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