Let's take a group G with order 45. Then |G|=45=32⋅5.

Find Sylow groups

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.

Learn how to find the Sylow 3-subgroups and Sylow 5-subgroups in a group of order 45 using Sylow's theorems. Understand the structure and implications of these subgroups, including potential group structures and subgroup interactions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation