Suppose that a is a 6 cycle and b is a 5 cycle. Determine whether a^5b^4a(-1)b^(-3) a^5 is even or odd

To determine whether the given permutation expression \( a^5 b^4 a^{-1} b^{-3} a^5 \) is even or odd, we need to consider the nature of the permutations \( a \) and \( b \).

### Step 1: Understanding the Cycles
- \( a \) is a 6-cycle: A 6-cycle permutation has an odd number of transpositions (i.e., 5 transpositions) and is therefore an **odd permutation**.
- \( b \) is a 5-cycle: A 5-cycle permutation has an even number of transpositions (i.e., 4 transpositions) and is therefore an **odd permutation**.

### Step 2: Properties of Permutations
- The power of an odd permutation (like \( a^5 \)) will remain odd.
- The inverse of an odd permutation (like \( a^{-1} \)) will also be odd.
- The product of an odd number of odd permutations is odd.
- The product of an even number of odd permutations is even.

### Step 3: Analyzing the Expression
Let's break down the permutation expression \( a^5 b^4 a^{-1} b^{-3} a^5 \):

1. \( a^5 \): Since \( a \) is odd, \( a^5 \) is odd.
2. \( b^4 \): Since \( b \) is odd, \( b^4 \) is even (because any odd permutation raised to an even power is even).
3. \( a^{-1} \): Since \( a \) is odd, \( a^{-1} \) is odd.
4. \( b^{-3} \): Since \( b^{-1} \) is odd and \( b^{-3} = (b^{-1})^3 \), \( b^{-3} \) is odd.
5. \( a^5 \) (again): As before, \( a^5 \) is odd.

Now, we multiply them together:

- \( a^5 \) is odd.
- \( b^4 \) is even.
- \( a^{-1} \) is odd.
- \( b^{-3} \) is odd.
- \( a^5 \) is odd.

So, we have:
\[
(\text{odd}) \times (\text{even}) \times (\text{odd}) \times (\text{odd}) \times (\text{odd})
\]

### Step 4: Conclusion
- Odd × Even = Even
- Even × Odd = Even
- Even × Odd = Even

The entire product is **even**.

So, the permutation \( a^5 b^4 a^{-1} b^{-3} a^5 \) is an **even** permutation.

Would you like any more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How does the parity of permutations affect the composition of two cycles?
2. What happens to the parity of a permutation when it is squared?
3. Can a permutation be both even and odd? Why or why not?
4. How do you determine the parity of a composition of multiple cycles?
5. What is the significance of the sign of a permutation in group theory?

**Tip:** Remember that the product of an even number of odd permutations is always even!

Explore the analysis of the permutation expression a^5 b^4 a^(-1) b^(-3) a^5 to determine its parity (even or odd). Understand the concept of permutation cycles, transpositions, and the product of permutations. Suitable for advanced high school students and beyond.

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