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.
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.
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A subgroup of the symmetric group of a set, acting faithfully by permutations. A finite permutation group specified by generators can be enumerated by repeatedly multiplying the elements already found by those generators until no new element appears. Its orbits and conjugacy classes can then be computed directly.
A permutation group is a mathematical concept in group theory that consists of a set of permutations of a given set combined with the operation of composition. Here's a more detailed breakdown of the concept: 1. **Permutations**: A permutation of a set is a rearrangement of its elements. For a finite set with \( n \) elements, a permutation is simply a bijective function from the set to itself.