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by Ciro Santilli (@cirosantilli, 37)

Stabilizer (group)

 ... Algebra Group Important mathematical group Important discrete mathematical group Permutation Permutation group
 0 By others on same topic  0 Discussions  Updated 2025-05-26  +Created 1970-01-01  See my version
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.
Bibliography:
  • mathworld.wolfram.com/Stabilizer.html
  • ncatlab.org/nlab/show/stabilizer+group from NLab

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