To analyze a group $G$ of order 45 and determine its Sylow subgroups, we will apply Sylow's theorems to identify the number and structure of these subgroups. The order of $G$ is $|G| = 45 = 3^2 \times 5$.

Step-by-Step Analysis

Sylow 3-subgroups

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

   Sylow's theorems state that the number $n_3$ of Sylow 3-subgroups satisfies:

   • $n_3 \equiv 1 \pmod{3}$
   • $n_3 \mid 45 / 9 = 5$

   The divisors of 5 are 1 and 5. Thus, $n_3$ can be 1 or 5.

   • If $n_3 = 1$, there is a unique Sylow 3-subgroup, which is normal in $G$.

2. Structure of the Sylow 3-subgroups:

   Each Sylow 3-subgroup has order $9$. The possible structures for groups of order 9 are:

   • Cyclic group $\mathbb{Z}_9$
   • Direct product of cyclic groups $\mathbb{Z}_3 \times \mathbb{Z}_3$

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. Thus, $n_5$ can be 1, 3, or 9.

   • If $n_5 = 1$, there is a unique Sylow 5-subgroup, which is normal in $G$.

2. Structure of the Sylow 5-subgroups:

   Each Sylow 5-subgroup has order 5. The only group of order 5 is:

   • Cyclic group $\mathbb{Z}_5$

Possible Structures of $G$

1. Case 1: $n_3 = 1$ and $n_5 = 1$

   • Both Sylow subgroups are unique and normal. In this case, $G$ is the direct product of its Sylow subgroups: $G \cong \mathbb{Z}_9 \times \mathbb{Z}_5$

2. Case 2: $n_3 = 5$ and/or $n_5 > 1$

   • If $n_3 = 5$ or $n_5 > 1$, the group $G$ could be non-abelian. The interaction between the Sylow subgroups must be further examined.

Conclusion

Given that $|G| = 45$ and the constraints on Sylow subgroups, if both $n_3 = 1$ and $n_5 = 1$, $G$ is likely to be abelian and isomorphic to $\mathbb{Z}_9 \times \mathbb{Z}_5$.

If either $n_3 = 5$ or $n_5 = 3$, $G$ might have a more complex structure, potentially involving non-abelian characteristics, such as a semidirect product.

Further Exploration

1. What implications do normal Sylow subgroups have on the structure of $G$?
2. How can we determine if $G$ is a semidirect product in the non-abelian case?
3. What are the possible actions of a Sylow 5-subgroup on a Sylow 3-subgroup?
4. Can $G$ have a subgroup of order 15, and what would it imply?
5. How does the existence of multiple Sylow 3-subgroups affect the normality of subgroups in $G$?
6. What are the properties of the direct product $\mathbb{Z}_9 \times \mathbb{Z}_5$?
7. In which cases can $G$ be expressed as a nontrivial semidirect product?
8. If $G$ is non-abelian, how might the Sylow subgroups interact?

Tip: When examining groups with multiple Sylow subgroups, consider the potential for non-abelian structures, such as semidirect products, by investigating possible subgroup interactions.