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.