Question

Write $\left(12\right)\left(34\right)$ as the product of two 3-cycles

Short answer

The product of two 3-cycles is$\left(142\right)\left(143\right)$

Step-by-step solution

Important theorem

Theorem 7.26: Every permutation in ${\mathit{S}}_{\mathbf{n}}$ is a product of (not necessarily disjoint) transpositions.

Writing (12)(34) as a product of two 3-cycles.

We know any permutation can be given as a product of transposition.

From computation,

.$\left(12\right)\left(34\right)=\left(142\right)\left(143\right)$

The product of two 3-cycles is$\left(142\right)\left(143\right)$