Question

List all the 3-cycles in ${\mathit{S}}_{\mathbf{4}}$ .

Short answer

The set of 3-cycles in$S_4$ are $\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(234\right),\left(243\right),\left(134\right),\left(143\right)\}$.

Step-by-step solution

Determine that 3-cycles in S4

Consider that$S_4$ has 3-cycles.

The set of all 3-cycles are in$S_4$ are follows:

$\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(234\right),\left(243\right),\left(134\right),\left(143\right)\}$

Thus, the set of 3-cycles in$S_4$ are$\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(234\right),\left(243\right),\left(134\right),\left(143\right)\}$ .