Let's take a group G with order 45. Then |G|=45=3^2⋅5. With sylow

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.

Explore the analysis of Sylow subgroups in a group of order 45 using Sylow's theorems. Understand the structure and number of Sylow 3-subgroups and Sylow 5-subgroups in detail.

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