Determine whether $(2 3 4 5)(2 4 6 7)$ is even or odd.

Question

Is the following permutation is even or odd? $$(2 3 4 5)(2 4 6 7).$$

I have the feeling it is even. But how to explain it?

Answer 1

Whatever parity a cycle of length $4$ has, you have composed two of them. Your permutation must therefore necessarily be even.

Answer 2

$$(2 3 4 5)\cdot(2 4 6 7)=(24)(32)(54)\cdot (26)(24)(67)$$ So it is an even permutation as it is product of $6$ (an even number) transpositions..

Answer 3

If we are dealing with the composition of two cycles that both have length $n$ then both can be written as a composition of $n-1$ transpositions.

So the total number of transpositions is $2n-2$ hence is even, and that is enough to conclude that the composition is an even permutation.