If $X$ is an infinite set, then the finitary alternating group on $X$ can be defined in the following equivalent ways: 1. the group of all even permutations on $X$ under composition; 2. the kernel of the sign homomorphism on the finatary symmetric group on $X$.
First, what does even permutation mean if $X$ is infinite? Second, how does one construct the homomorphism mentioned in statement 2.?