$(1\ 3\ 5\ 4)(2\ 6)$ is both even and odd, how is this possible?

Question

Is the permutation $(1\ 3\ 5\ 4)(2\ 6)$ even or odd?

This permutation has order $4$ ($\operatorname{lcm}(4,2)=4$), so it must be an odd permutation. But $(1\ 3\ 5\ 4)(2\ 6) = (1\ 4)(1\ 5)(1\ 3)(2\ 6)$, so it is an even permutation. How is this possible?

Answer 1

If a cycle has even/odd order (or equivalently, length), then it is an odd/even permutation. But $(1\ 3\ 5\ 4)(2\ 6)$ is not a cycle, it is a product of two cycles, so the result doesn't apply.

As you demonstrated, $(1\ 3\ 5\ 4)(2\ 6)$ is a product of an even number of transpositions and is therefore an even permutation.

Answer 2

A permutation is even when it can be written as an even number of transpositions, i.e., an even number of two cycles.