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To solve the permutation \( \sigma = \left( \begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 1 & 6 & 5 \end{array} \right) \) as a product of 2-cycles (transpositions), we need to decompose it into cycles and then convert each cycle into 2-cycles.

First, let's write \( \sigma \) in cycle notation:
1. \(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1\), which gives us the cycle \((1\ 2\ 3\ 4)\).
2. \(5 \rightarrow 6 \rightarrow 5\), which gives us the cycle \((5\ 6)\).

So, \( \sigma = (1\ 2\ 3\ 4)(5\ 6) \).

Next, we convert each cycle into 2-cycles:
1. The cycle \((1\ 2\ 3\ 4)\) can be written as:
   \[
   (1\ 4)(1\ 3)(1\ 2)
   \]

2. The cycle \((5\ 6)\) is already a 2-cycle:
   \[
   (5\ 6)
   \]

Therefore, the permutation \( \sigma \) can be expressed as a product of 2-cycles:
\[
\sigma = (1\ 4)(1\ 3)(1\ 2)(5\ 6)
\]

Would you like further details or have any questions?

Here are some questions you might find interesting:
1. How do you convert a general cycle into 2-cycles?
2. What is the significance of expressing permutations as products of transpositions?
3. Can you provide more examples of permutations and their 2-cycle decompositions?
4. How do transpositions relate to the concept of even and odd permutations?
5. How can we use permutations in practical applications?

**Tip:** When decomposing cycles into 2-cycles, start from the first element and pair it with each subsequent element in the cycle until you reach the second to last element.

Learn how to decompose a permutation into 2-cycles (transpositions) using cycle notation. This problem involves converting cycles into their equivalent 2-cycles and understanding the significance of expressing permutations in this form.

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