If $X$ is an infinite set, then the finitary alternating group on $X$ can be defined in the following equivalent ways:
- the group of all even permutations on $X$ under composition
- the kernel of the sign homomorphism on the finitary symmetric group on $X$. (see here)
First, what does even permutation mean if $X$ is infinite? Second, how does one construct the homomorphism mentioned in statement 2.? For example, if $X = \{1,...,n\} \times \Bbb{N}$, what does finitary alternating symmmetric mean in this case?
The finitary symmetric group is the group of all permutations with finite support, or said in another way: An element in the finitary symmetric group of a set is a permtutation that permutes only finitely many elements of the set. You can then define the sign to be the sign of the permutation restricted to the support, which is a permutation of a finite set.
For $X= \{1,...,n\} \times \mathbb{N}.$ If $\tau$ is a finitary symmetric permuatation, then there is a finite subset $A$ of $X, $ such that for all $(a,b) \in X-A, \tau(a,b) = (a,b)$