Let σ = (24)(1759)(428)(735) ∈ S9.
1. Write σ as a product of disjoint cycles and why is it a even permutatation

Let's break down the permutation \( \sigma = (24)(1759)(428)(735) \) and write it as a product of disjoint cycles.

### Step 1: Apply the cycles one by one
- Start with the first cycle \( (24) \). This means:
  - \( 2 \) goes to \( 4 \) and
  - \( 4 \) goes to \( 2 \).
  
- Now move to the next cycle \( (1759) \). This means:
  - \( 1 \) goes to \( 7 \),
  - \( 7 \) goes to \( 5 \),
  - \( 5 \) goes to \( 9 \), and
  - \( 9 \) goes to \( 1 \).
  
- Next, consider \( (428) \). This means:
  - \( 4 \) goes to \( 2 \) (but \( 2 \) already goes to \( 4 \) from the first cycle, so \( 4 \) returns to \( 4 \)),
  - \( 2 \) goes to \( 8 \),
  - \( 8 \) goes to \( 4 \).
  
- Finally, \( (735) \). This means:
  - \( 7 \) goes to \( 3 \),
  - \( 3 \) goes to \( 5 \) (but \( 5 \) already goes to \( 9 \) from the second cycle),
  - \( 5 \) goes to \( 7 \).

### Step 2: Write the resulting disjoint cycles
To construct the disjoint cycles, we follow the path of each number:

- Start with \( 1 \): 
  - \( 1 \rightarrow 7 \rightarrow 3 \rightarrow 5 \rightarrow 9 \rightarrow 1 \).
  - This forms the cycle \( (17359) \).
  
- Start with \( 2 \): 
  - \( 2 \rightarrow 8 \rightarrow 4 \rightarrow 2 \).
  - This forms the cycle \( (284) \).

### Step 3: Express the permutation in disjoint cycles
The permutation \( \sigma \) can be written as:
\[ \sigma = (17359)(284) \]

### Step 4: Determine if the permutation is even or odd
To determine whether \( \sigma \) is an even or odd permutation, count the number of transpositions (2-cycles) in its disjoint cycle decomposition.

- The cycle \( (17359) \) has 5 elements, which corresponds to \( 5 - 1 = 4 \) transpositions.
- The cycle \( (284) \) has 3 elements, which corresponds to \( 3 - 1 = 2 \) transpositions.

Total number of transpositions: \( 4 + 2 = 6 \).

Since 6 is an even number, the permutation \( \sigma \) is an **even permutation**.

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Would you like more details on any step, or have any questions? Here are some related questions to explore further:

1. How do you decompose a general permutation into disjoint cycles?
2. What are transpositions, and how do they relate to the parity of a permutation?
3. Why does the product of an even number of transpositions result in an even permutation?
4. Can you provide an example of an odd permutation and explain why it's odd?
5. How do you calculate the order of a permutation given its disjoint cycle form?

**Tip:** Remember that the parity (even or odd) of a permutation is determined by the number of transpositions in its decomposition. If it's an even number, the permutation is even; if it's odd, the permutation is odd.

Understand how to decompose the permutation σ = (24)(1759)(428)(735) into disjoint cycles and determine its parity (even or odd). This problem involves key concepts in permutations, including disjoint cycles and transpositions. Suitable for advanced high school students and those interested in abstract algebra.

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