Finitary symmetric group is normal in symmetric group
From Groupprops
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
.