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A stabilizer is a subgroup of a group that keeps certain elements unchanged during a group action. It identifies the elements of the group that act as symmetries for a given object or set, helping to understand the structure and properties of the group through its action on various sets.
Stabilizers turn out to be a subgroup! The orbit stabilizer theorem states that the product of the number of threads which map an element into itself (size of stabilizer set) and number of threads which push that same element into different elements (orbit) equals the order of the original group!
Definition. In the context of algebraic groups and group actions, the stabilizer is a subgroup of a group that leaves a particular point or set invariant under the group action.
A stabilizer is a subgroup of a group that keeps a particular element fixed under the action of that group. In other words, it consists of all the elements in the group that, when applied to a specific element, do not change that element.
26 lis 2024 · Let G be a permutation group on a set Omega and x be an element of Omega. Then G_x={g in G:g(x)=x} (1) is called the stabilizer of x and consists of all the permutations of G that produce group fixed points in x, i.e., that send x to itself.
The stabilizer of \(s\) is the set \(G_s = \{g\in G \mid g\cdot s=s \}\), the set of elements of \(G\) which leave \(s\) unchanged under the action. For example, the stabilizer of the coin with heads (or tails) up is \(A_n\), the set of permutations with positive sign.
The stabilizers for any two elements in the same orbit are isomorphic, via an inner automorphism on G. Conversely, the a-conjugate of the stabilizing subgroup of x stabilizes another element in the orbit of x, namely a(x).
