Definition. normalizer of a subgroup [fgap-000F]
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July 28, 2026

Let \(N\) be a subgroup of \(G\). The normalizer of \(N\) in \(G\) is \[ N_G(N)=\{g\in G:gNg^{-1}=N\}. \] It is the largest subgroup of \(G\) in which \(N\) is normal. In particular, a subgroup \(H\leq G\) acts on \(N\) by conjugation precisely when \[ H\leq N_G(N). \] See [fre2023discrete, sec. 4.2.6, pp. 56--57].

The ambient group matters. If \(H\) normalizes \(N\), then \(N\) is normal in the subgroup generated by \(N\) and \(H\). This does not imply that \(N\) is normal in all of \(G\).

For the quaternion subgroups in Quaternion and order-three subgroups, the three conjugation calculations in the Hurwitz action on the quaternion group give \[ C\leq N_{\mathbb {H}^{\times }}(Q). \] They do not claim that \(Q\) is normal in \(\mathbb {H}^{\times }\).

Definition. normalizer of a subgroup [fgap-000F]
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Reference. Discrete, finite and Lie groups: comprehensive group theory in geometry and analysis [fre2023discrete]

From the Hurwitz action to the binary tetrahedral group [fgap-000E]

Definition. normalizer of a subgroup [fgap-000F]
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Lemma. a normalized product is the generated subgroup [fgap-000G]

Proof.

Definition. complementary subgroups [fgap-000H]

Theorem. the internal semidirect-product theorem [fgap-000I]

Proof.

Example. the binary tetrahedral subgroup of the quaternions [fgap-000J]

Example. the twenty-four Hurwitz units [fgap-000K]

Remark. internal, external, and abstract models [fgap-000L]

Definition. the distinguished involution of abstract 2T [fgap-0014]

Quaternion and order-three subgroups [fgap-0007]

Definition. group action [fgap-0008]

Definition. the quaternion group inside Hamilton's Quaternions [fgap-0009]

Lemma. normal forms in the quaternion subgroup [fgap-000A]

Proof.

Lemma. the cyclic subgroup generated by the Hurwitz unit [fgap-000B]

Proof.

Example. the Hurwitz action on the quaternion group [fgap-000C]

Remark. concrete and abstract factor models [fgap-000D]

Example. the Hurwitz action on the quaternion group [fgap-000C]