Ciro Santilli Suppose we have a given permutation group that acts on a set of n elements.
If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.
Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since all groups are isomorphic to a subgroup of the symmetric group
TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.
Group of all permutations.
Or in other words: symmetric groups are boring, because they are basically everything already!
Group of even permutations.
Note that odd permutations don't form a subgroup of the symmetric group like the even permutations do, because the composition of two odd permutations is an even permutation.
www.youtube.com/watch?v=U_618kB6P1Q GT18.2. A_n is Simple (n ge 5) by MathDoctorBob (2012)