Stabilizer subgroup¶
For a group G acting on a set X, the stabilizer subgroup of a point x is the subgroup of exactly those elements of G that leave x fixed under the action.
Core Idea¶
For a group G acting on a set X, the stabilizer subgroup of a point x, written Gx or StabG(x), is {g in G | g·x = x}. Closure, identity, and inverses follow from the group action, so the fixers of x form a subgroup rather than merely a set of transformations. The identity is relative to both the action and the selected point. The same abstract group can have different stabilizers under different actions, and different points in one action can have different stabilizers.
Scope of Application¶
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DefinitionLeft group action. If G is a group with identity element e , and X is a set, then a (left) group action \alpha of G on X is a function.
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DefinitionLeft group action. The second axiom states that the function composition is compatible with the group multiplication; they form a commutative diagram.
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DefinitionLeft group action. From these two axioms, it follows that for any fixed g in G , the function from X to itself which maps x to g\cdot x is a bijection, with inverse.
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Right group action. Likewise, a right group action of G on X is a function.
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Orbits and stabilizers. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.
Clarity¶
A clear use of Stabilizer subgroup names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is For a group G acting on a set X, the stabilizer subgroup of a point x is the subgroup of exactly those elements of G that leave x fixed under the action.
Manages Complexity¶
Stabilizer subgroup compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—one also sees that consists only of the identity automorphism, as any element of fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3.—and the practical consequence—for example, the action of any group on itself by left.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: For a group G acting on a set X, the stabilizer subgroup of a point x is the subgroup of exactly those elements of G that leave x fixed under the action.
- Check operation and conditions. The quaternions with norm 1 (the versors), as a multiplicative group, act on : for any such quaternion , the mapping is a counterclockwise rotation through an angle about an axis given.
Knowledge Transfer¶
Within the home domain. Knowledge about Stabilizer subgroup transfers literally when a new case preserves the same carrier type, relation, and recognition test. If G is a group with identity element e , and X is a set, then a (left) group action \alpha of G on X is a function. The second axiom states that the function composition is compatible with the group multiplication; they form a commutative diagram. Beyond the home domain. No canonical parent is asserted for Stabilizer subgroup.
Relationships to Other Abstractions¶
Current abstraction Stabilizer subgroup Domain-specific
Parents (1) — more general patterns this builds on
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Stabilizer subgroup is a kind of Group Prime
Stabilizer subgroup is a domain-specific kind of group under the frozen identity and differentia.
Hierarchy paths (5) — routes to 5 parentless roots
- Stabilizer subgroup → Group → Monoid → Semigroup → Set and Membership
- Stabilizer subgroup → Group → Monoid → Identity Element
- Stabilizer subgroup → Group → Monoid → Semigroup → Closure
- Stabilizer subgroup → Group → Monoid → Semigroup → Associativity → Invariance
- Stabilizer subgroup → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Stabilizer subgroup sits in a sparse region of the domain-specific corpus (72nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Convergence Group — 0.84
- Complex representation — 0.84
- Wandering set — 0.83
- Additive group — 0.83
- Complex conjugate representation — 0.83
Computed from structural-signature embeddings · 2026-10-08