Question

Let $\mathbf{\sigma }$ be a product of disjoint cycles of the same length. Prove that $\mathit{\sigma }$ is a power of a cycle.

Short answer

It is proved that $\sigma$ is a power of a cycle.

Step-by-step solution

The disjoint cycle σ

Let $(a_{11}{a}_{21}\dots a_{n1})(a_{12}{a}_{22}\dots a_{n2})\dots (a_{1m}{a}_{2m}\dots a_{nm})$ be the disjoint cycles.

Then $\sigma =(a_{11}{a}_{21}\dots a_{n1})(a_{12}{a}_{22}\dots a_{n2})\dots (a_{1m}{a}_{2m}\dots a_{nm})$ be the product of m disjoint n-cycles.

σ is a power of a cycle

Let $\tau =(a_{11}{a}_{12}\dots a_{1m}{a}_{21}{a}_{22}\dots a_{2m}\dots a_{n1}{a}_{n2}\dots a_{nm})$ be the nm-cycle which can be written as ${\tau }^n=\sigma$.

Thus, $\sigma$ can be written as a power of nm-cycle.

Therefore, if $\sigma$ is the product of disjoint cycles of the same length then $\sigma$ is a power of a cycle.