Question

Let $\mathbf{\sigma }$ is a k-cycle in ${\mathbf{S}}_n$.

1. Prove that ${\mathbf{\sigma }}^2$ is a cycle if and only if k is odd.

Short answer

It is proved that ${\mathrm{\sigma }}^2$ is a cycle if and only if k is odd.

Step-by-step solution

Definition of k-cycle

Consider that ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots ,{\mathrm{a}}_{\mathrm{k}}$ (with $k\ge 1$) as distinct elements of the set $\{1,2,3,\dots ,n\}$. Then $\{a_1{a}_2{a}_3\dots a_k\}$ represents the permutation in ${\mathrm{S}}_{\mathrm{n}}$ which maps $a_1$ to role="math" localid="1654330178252" $a_2,a_2$ to $a_3,\dots ,a_{k-1}$ to $a_k$, and$a_k\mathrm{to}{a}_1$ , and maps all other elements of $\{1,2,3,\dots .n\}$ to itself. $(a_1,a_2,a_3,\dots ,a_k)$ is known as the cycle of length$k$ or $\mathbf{k}\mathbf{-}\mathbf{cycle}$.

Show that σ2  is a cycle if and only if k is odd

Consider $\sigma =(a_1\dots a_k)\in S_n$ as a role="math" localid="1654330537967" $k$ cycle. When role="math" localid="1654330542004" $k$ is odd, there is ${\sigma }^2=(a_1{a}_3\dots a_k{a}_2{a}_4\dots a_{k-1})$, a role="math" localid="1654330534232" $k$-cycle.

Now, assume that $k$is even, then ${\sigma }^2=(a_1{a}_3\dots a_{k-1})(a_2{a}_4\dots a_k)$, a product of disjoint $\frac{k}{2}$cycles.

Hence, it is proved that ${\sigma }^2$ is a cycle if and only if k is odd.