An $m$-cycle is odd iff $m$ is even

Show that an $m$-cycle is an odd permutation iff $m$ is even.

Solution

An $m$-cycle can be written as a product of $m-1$ transpositions: \[ (a_1\ a_2\ \cdots\ a_m) = (a_1\ a_m)(a_1\ a_{m-1})\cdots(a_1\ a_2). \] Therefore its sign is \[ (-1)^{m-1}. \] This equals $-1$ exactly when $m-1$ is odd, equivalently when $m$ is even. Hence an $m$-cycle is odd iff $m$ is even.