# Difference between revisions of "Finitary symmetric group is normal in symmetric group"

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Let <math>S</math> be a set. Let <math>K = \operatorname{Sym}(S)</math> be the [[symmetric group]] on <math>S</math>, and <math>G</math> be the [[finitary symmetric group]] on <math>S</math>. In other words, <math>G</math> is the subgroup of <math>K</math> comprising the finitary permutations, i.e., the permutations that move only finitely many elements. Then, <math>G</math> is a [[normal subgroup]] of <math>K</math>. | Let <math>S</math> be a set. Let <math>K = \operatorname{Sym}(S)</math> be the [[symmetric group]] on <math>S</math>, and <math>G</math> be the [[finitary symmetric group]] on <math>S</math>. In other words, <math>G</math> is the subgroup of <math>K</math> comprising the finitary permutations, i.e., the permutations that move only finitely many elements. Then, <math>G</math> is a [[normal subgroup]] of <math>K</math>. | ||

+ | |||

+ | ==Related facts== | ||

+ | |||

+ | * [[Finitary symmetric group is characteristic in symmetric group]] | ||

+ | * [[Finitary symmetric group is automorphism-faithful in symmetric group]] | ||

==Proof== | ==Proof== |

## Latest revision as of 18:41, 5 April 2009

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Finitary symmetric group (?)) satisfying a particular subgroup property (namely, Normal subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

## Statement

Let be a set. Let be the symmetric group on , and be the finitary symmetric group on . In other words, is the subgroup of comprising the finitary permutations, i.e., the permutations that move only finitely many elements. Then, is a normal subgroup of .

## Related facts

- Finitary symmetric group is characteristic in symmetric group
- Finitary symmetric group is automorphism-faithful in symmetric group

## Proof

**Given**: A set . is a subgroup of . and .

**To prove**: .

**Proof**: If and , we have:

.

Thus, is moved by if and is moved by . Since is a permutation, this shows that the number of points moved by and is equal. In particular, also moves only finitely many points, and hence is in .

Thus, is normal in .