Question

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

(b) If$\mathbf{k}\mathbf{=}\mathbf{2}\mathbf{t}$ , prove that there are t-cyclesrole="math" localid="1654330901733" $\mathit{\tau }$ and$\mathit{\beta }$ such that ${\mathbf{\sigma }}^2\mathbf{=}\tau \mathbf{\beta }$.

Short answer

b) It is proved that there are $t$-cycle $\tau$ and $\beta$ such that ${\sigma }^2=\tau \beta$.

Step-by-step solution

Statement of Exercise 7.27

Exercise 7.27states that when $\sigma$ is a k-cyclefor $k$ odd, then there would be cycle $\tau$ such that ${\tau }^2=\sigma$.

Show that there are t-cycles  src="data:image/svg+xml;base64,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" role="math" localid="1654331265589" τ and  β such that  σ2=τβ

When $k=2t$ is even then $r\equiv 0\left(\mathrm{mod}t\right)$and the smallest positive solution would be $t$. Then ${\sigma }^2$ has a $t-$cycle starting with $a_0$. Starting with any $a_j$ argument would holds: every cycle in the decomposition of ${\sigma }^2$would be a $t-$cycle. Then, the product of two disjoint $t-$cycles would be a role="math" localid="1654331505329" ${\sigma }^2$.

In general, the permutation ${\sigma }^m$for every role="math" localid="1654331660314" $m$will be the product of $k/d$ $d-$cycles, where $d=(m,k)$.

Hence, it is proved that there are $t$-cycle $\tau$ and $\beta$ such that ${\sigma }^2=\tau \beta$.