(12345) is an even permutation of S_5. True or False?

Question

The answer i had for this question was True, yet i'm not sure. Well, from what I know so far was that: $(12345)$ can be expressed as a number of 4 transpositions such as: $(12)(23)(34)(45)$ which is of course even, so I concluded that the statement was True. I also recall that cycles of odd length can only be expressed as even permutations. Is this true? Like I said, I'm not totally sure about the claims that I've made and it would be well-appreciated if someone shed some light. Thank you! :)

Answer 1

In general, if a $k$ is odd, then any $k$-cycle can be decomposed into an even number of transpositions. A simple algorithm for decomposing any cycle (1 2 3 ... n) into a product of transpositions is as follows:

(1 n)(1 n-1)....(1 3)(1 2)

Answer 2

For S_5, one can tell if a permutation is even or odd thus.

Write the known permutation around as a pentagon vertices. Now, draw a line from the second permutation, and connect the first to the last.

If the line crosses an odd number of times (1, 5) then the permutation is odd. If it crosses an even number of times (0, 2), it is even.