Lemma. a normalized product is the generated subgroup [fgap-000G]
✍️source

July 28, 2026

Let \(N,H\leq G\). If \(H\leq N_G(N)\), then the set product \[ NH=\{nh:n\in N,\ h\in H\} \] is a subgroup and \[ N\vee H=NH. \] Moreover, \(N\) is normal in \(N\vee H\).

Proof.

July 28, 2026

Normalization gives the calculation needed for closure: \[ (n_1h_1)(n_2h_2) =n_1(h_1n_2h_1^{-1})h_1h_2\in NH. \] It also gives \((nh)^{-1}=(h^{-1}n^{-1}h)h^{-1}\in NH\). Thus \(NH\) is a subgroup containing both \(N\) and \(H\), while every subgroup containing \(N\) and \(H\) contains every product \(nh\). This proves \(N\vee H=NH\). Finally, conjugation by generators from \(N\) and \(H\) preserves \(N\), so \(N\) is normal in their generated subgroup.

This is the small bridge between an action by conjugation and an internal product. Compare the normalizer and internal-product discussion in [fre2023discrete, secs. 4.2.6 and 4.2.13, pp. 56--57 and 62--63].

Lemma. a normalized product is the generated subgroup [fgap-000G]
✍️source

Proof.

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]

Lemma. a normalized product is the generated subgroup [fgap-000G]
✍️source

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]

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