To find the Sylow groups of a group $G$ with order $|G| = 45 = 3^2 \cdot 5$, we need to determine the number and structure of the Sylow $p$-subgroups for each prime divisor of 45.

Step-by-Step Solution

Sylow 3-subgroups

1. Determine the number of Sylow 3-subgroups, $n_3$:

   By Sylow's theorems, the number of Sylow $p$-subgroups, $n_p$, satisfies two conditions:

   • $n_3 \equiv 1 \pmod{3}$
   • $n_3 \mid 45 / 3^2 = 5$

   The divisors of 5 are 1 and 5. Therefore, $n_3$ must be either 1 or 5.

   If $n_3 = 1$, there is a unique Sylow 3-subgroup, and it must be normal in $G$.

2. Structure of the Sylow 3-subgroups:

   Each Sylow 3-subgroup has order $3^2 = 9$. The only group of order 9 up to isomorphism is the cyclic group $\mathbb{Z}_9$ or the direct product of two cyclic groups of order 3, $\mathbb{Z}_3 \times \mathbb{Z}_3$.

Sylow 5-subgroups

1. Determine the number of Sylow 5-subgroups, $n_5$:

   Similarly, for Sylow 5-subgroups, the conditions are:

   • $n_5 \equiv 1 \pmod{5}$
   • $n_5 \mid 45 / 5 = 9$

   The divisors of 9 are 1, 3, and 9. Therefore, $n_5$ must be 1, 3, or 9.

   If $n_5 = 1$, there is a unique Sylow 5-subgroup, and it must be normal in $G$.

2. Structure of the Sylow 5-subgroups:

   Each Sylow 5-subgroup has order 5. The only group of order 5 is the cyclic group $\mathbb{Z}_5$.

Conclusion

The Sylow 3-subgroups and Sylow 5-subgroups are determined as follows:

• Sylow 3-subgroups:

  • Possible number: $n_3 = 1$ or 5.
  • Structure: Cyclic group $\mathbb{Z}_9$ or the elementary abelian group $\mathbb{Z}_3 \times \mathbb{Z}_3$.

• Sylow 5-subgroups:

  • Possible number: $n_5 = 1$, 3, or 9.
  • Structure: Cyclic group $\mathbb{Z}_5$.

Additional Analysis

If $n_3 = 1$ and $n_5 = 1$, both Sylow subgroups are normal, making $G$ a direct product $G \cong \mathbb{Z}_9 \times \mathbb{Z}_5$.

If neither of the Sylow numbers is 1, the group might be more complex, potentially non-abelian. Further investigation would be necessary to identify specific group properties or the existence of additional structures like semidirect products.

Further Exploration

1. What is the significance of a Sylow subgroup being normal in $G$?
2. Can $G$ be non-abelian if both Sylow subgroups are normal?
3. How does the Sylow counting theorem help determine group structure?
4. Can there be any non-trivial homomorphisms between the Sylow subgroups in $G$?
5. If $n_3 = 5$ and $n_5 = 3$, what are the implications for the structure of $G$?
6. What are the possible actions of Sylow 5-subgroups on Sylow 3-subgroups?
7. How does the existence of unique Sylow subgroups affect the group homomorphisms?
8. In which cases could $G$ be isomorphic to $\mathbb{Z}_9 \times \mathbb{Z}_5$?

Tip: When analyzing the structure of finite groups, always check the conditions given by Sylow's theorems to gain insight into possible group structures and subgroup interactions.