Finitary symmetric group is normal in symmetric group
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 (?)).
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 .
- Finitary symmetric group is characteristic in symmetric group
- Finitary symmetric group is automorphism-faithful in symmetric group
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 .