Question

Let $\mathbf{\sigma }$and $\mathbf{\pi }$be transposition in ${\mathbf{S}}_{\mathbf{n}}$with $\mathbf{n}\mathbf{\ge }\mathbf{3}$. Prove that $\mathbf{\alpha \pi }$is a product of (not necessarily disjoint) 3-cycles.

Short answer

Answer:

It is proved that $\alpha \pi$ is a product of (not necessarily disjoint) 3-cycles.

Step-by-step solution

Referring to Theorem 7.26

Theorem 7.26

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

Given that $\alpha$and$\pi$are transpositions in $S_n$with $n\ge 3$.

Proving that απ is a product of (not necessarily disjoint) 3-cycles

According to the question, consider $\alpha =\left(1234\right)$and $\pi =\left(243\right)$.

Therefore, find $\alpha \pi$as:

$$\begin{gathered} \alpha \pi =\left(1234\right)\left(243\right) \\ =\left(23\right)\left(34\right)\left(14\right) \end{gathered}$$

Since the result is a product of 3-cycles and from above result, it is proved that $\alpha \pi$is a product of (not necessarily disjoint) 3-cycles.

Hence, $\alpha \pi$is a product of (not necessarily disjoint) 3-cycles.