Exercise
Which of the following permutations in S4 is even?
a = (1,2,3,4).
b = (1,2)(3,4)(1,2,3)(2,4).
c = (1,2,3,4)(1,2).
We write (1,2,3,4) = (1,4)(1,3)(1,2) and conclude that a is an odd permutation.
We write (1,2)(3,4)(1,2,3)(2,4) = (1,4)(1,2)(1,3) and conclude that b is an odd permutation.
The permutation c is even, because c = (1,4)(1,3)
is the product of an even number of transpositions.