LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

If $y = g \cdot x$ , then $G_{y} = g G_{x} g^{- 1}$

Statement

Let $G$ act on $X$ . If $y = g \cdot x$ , then

$G_{y} = g G_{x} g^{- 1} .$

In particular, stabilizers of points in the same orbit are conjugate and hence isomorphic.

Facts & Assumptions

Given: A left action of $G$ on $X$ , points $x, y \in X$ , and $g \in G$ with $y = g \cdot x$ .

[L1]

The stabilizer is $G_{x} = {h \in G : h \cdot x = x}$ (The orbit $G \cdot x$ and stabilizer $G_{x}$ of a point in a group action).

[L2]

A left action satisfies $(a b) \cdot z = a \cdot (b \cdot z)$ and $e \cdot z = z$ (Left group actions, transitive actions, and faithful actions).

[L3]

Conjugation $h \mapsto g h g^{- 1}$ is an automorphism of $G$ (Conjugation $x \mapsto g x g^{- 1}$ is an automorphism).

Proof

technique · direct

1.1

For $h \in G$ , one has $h \in G_{y}$ exactly when $h \cdot (g \cdot x) = g \cdot x$ , which by [L2] is equivalent, after applying $g^{- 1}$ , to $(g^{- 1} h g) \cdot x = x$ , that is, to $g^{- 1} h g \in G_{x}$ .

L1L2L3

2.1

The last condition is equivalent to $h \in g G_{x} g^{- 1}$ , so $G_{y} = g G_{x} g^{- 1}$ ; [L3] also makes conjugation an isomorphism from $G_{x}$ onto $G_{y}$ .

step 1.1L3algebra∎

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets (standard reference, not scraped)

If y=g⋅xy=g⋅x, then Gy=gGxg−1Gy​=gGx​g−1