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.