TheoremStatement: 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.

The conjugates of $H$ are in bijection with $G / N_{G} (H)$ and, for finite $G$ , number $[G : N_{G} (H)]$

Statement

Let $H \leq G$ . The rule

$G / N_{G} (H) ⟶ {g H g^{- 1} : g \in G}, g N_{G} (H) ⟼ g H g^{- 1},$

is a well-defined bijection. If $G$ is finite, the number of distinct conjugates of $H$ is $[G : N_{G} (H)]$ .

Facts & Assumptions

Given: A group $G$ and a subgroup $H \leq G$ .

[L1]

Orbit-stabiliser identifies an orbit with the cosets of its stabilizer (Orbit-stabiliser: $G / G_{x} \to G \cdot x$ , $g G_{x} \mapsto g \cdot x$ , is a well-defined bijection).

[L2]

The cardinality of a finite orbit is the index of its stabilizer (Orbit-stabiliser cardinality: $∣ G \cdot x ∣ = [G : G_{x}]$ whenever either side is finite, and $∣ G ∣ = ∣ G_{x} ∣ ∣ G \cdot x ∣$ for finite $G$ ).

[L3]

The normalizer is $N_{G} (H) = {g : g H g^{- 1} = H}$ (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup).

[L4]

The normalizer is a subgroup of $G$ ( $C_{G} (x)$ and $N_{G} (H)$ are subgroups of $G$ ).

[L5]

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

Proof

technique · direct

1.1

Let $G$ act on the set of subgroups of $G$ by $g \cdot K = g K g^{- 1}$ . By [L5], conjugation sends subgroups to subgroups, and the conjugation identities give the action laws.

2.1

The orbit of $H$ is its set of conjugates, while [L3] says that its stabilizer is $N_{G} (H)$ , a subgroup by [L4].

step 1.1L3L4

3.1

Applying [L1] gives the displayed bijection, and [L2] gives the finite count $[G : N_{G} (H)]$ .

step 2.1L1L2∎

Depends on

Used by

Dependency tree · two levels

18 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, Corollary 3.110 (standard reference, not scraped)

The conjugates of HH are in bijection with G/NG(H)G/NG​(H) and, for finite GG, number [G:NG(H)][G:NG​(H)]