Finitary alternating group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

For a finite set $S$, the finitary alternating group on $S$ is equal to the alternating group on $S$.

Let $S$ be an infinite set. The finitary alternating group on $S$ is defined in the following equivalent ways:

1. It is the group of all even permutations on $S$ under composition, that have finite support, viz permutations that fix all but finitely many elements.
2. It is the kernel of the sign homomorphism on the finitary symmetric group on $S$.

This contrasts the alternating group on $S$, which is defined as:

1. The group of all even permutations on $S$ under composition, including those with infinite support.
2. It is the kernel of the sign homomorphism on the symmetric group on $S$ (i.e. the symmetric group including permutations with infinite support).

Particular examples

• For any finite set $S$, the group is isomorphic to the alternating group of degree $|S|$.
• Finitary alternating group on the natural numbers