Finitary alternating group: Difference between revisions

Latest revision as of 17:06, 12 January 2024

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:

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.
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:

The group of all even permutations on $S$ under composition, including those with infinite support.
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

@@ Line 1: / Line 1: @@
-#redirect [[alternating group]]
+{{basicdef}}
+==Definition==
+For a finite set <math>S</math>, the finitary alternating group on <math>S</math> is equal to the [[alternating group]] on <math>S</math>.
+Let <math>S</math> be an infinite set. The '''finitary alternating group''' on <math>S</math> is defined in the following equivalent ways:
+# It is the group of all even permutations on <math>S</math> under composition, that have finite support, viz permutations that fix all but finitely many elements.
+# It is the kernel of the [[sign homomorphism]] on the [[finitary symmetric group]] on <math>S</math>.
+This contrasts the [[alternating group]] on <math>S</math>, which is defined as:
+# The group of all even permutations on <math>S</math> under composition, including those with infinite support.
+# It is the kernel of the [[sign homomorphism]] on the [[symmetric group]] on <math>S</math> (i.e. the symmetric group including permutations with infinite support).
+==Particular examples==
+* For any finite set <math>S</math>, the group is isomorphic to the alternating group of degree <math>|S|</math>.
+* [[Finitary alternating group on the natural numbers]]