Question

Prove that cycle $\left(a_1{a}_2....a_k\right)$ is even if only if k is odd.

Short answer

It is proved that cycle $\left({\mathrm{a}}_1{\mathrm{a}}_2.....{\mathrm{a}}_{\mathrm{k}}\right)$ is even if and only ifk is odd.

Step-by-step solution

Required Theorem

Theorem 7.26: Every permutation in S_n is a product of (not necessarily disjoint) transpositions.

Proving that cycle (a1a2.....ak)  is even only if k is odd

From the above theorem, we can write the given cycle as a product of its transpositions, i.e.,

$\left({\mathrm{a}}_1{\mathrm{a}}_2....{\mathrm{a}}_{\mathrm{k}}\right)=\left({\mathrm{a}}_1{\mathrm{a}}_2\right)\left({\mathrm{a}}_2{\mathrm{a}}_3\right)....\left({\mathrm{a}}_{\mathrm{k}-1}{\mathrm{a}}_{\mathrm{k}}\right)\mathrm{upto}\left(\mathrm{k}-1\right)\mathrm{terms}$

From the above expression, it can be seen that the product of transposition has k-1 terms.

Since the cycle is even, the terms in the transposition must be even.

Therefore, k-1 must be even, which implies that k must be odd. Hence, it is proved that the cycle $\left({\mathrm{a}}_1{\mathrm{a}}_2.....{\mathrm{a}}_{\mathrm{k}}\right)$is even if and only if k is odd.