Sufficient condition that a permutation is even

Question

Let $a$ be a permutation of $S_n$ and suppose that $a$ decomposes as a product

$a = \sigma_1 \circ \dots \circ \sigma_r \circ \gamma_1 \circ \dots \circ \gamma_s$

of disjoint cycles where $\sigma_i$ all have odd length and $\gamma_j$ all have even length. Give a necessary and sufficient condition on $r,s$ for $a$ to be an even permutation.

So for $a$ to be an even permutation, it should have odd length. Since $\gamma$ has even length, the product of those will have lcm of an even number. On the other hand, $\sigma$ has odd length so the product of those will have lcm of an odd number. Then the final product of $a$ will be a product between an even and an odd length, but I need to have $a$ as an odd length. And lcm of an even and an odd number is never odd... Did I miss something? Any help is appreciated.

Answer 1

Since $\operatorname{sgn}(a)=(-1)^r$, $a$ is an even permutation if and only if $r$ is even.

Answer 2

Try breaking it down like this: by definition a permutation is even if it can be written as a product of an even number of transpositions ($2$-cycles). So, each of the $\sigma_i$ can be written as a product of an even number of transpositions. And each of the $\gamma_i$ can be written as a product of an odd number of transpositions. To get the product to be a product of an even number of transpositions, it's necessary and sufficient that $s$ be even.