Definition

Direct and Semidirect Product

Lie Groups and Lie Algebras · lie-groups.tex

The direct product $G \times H$ of two groups (Definition~Group (Classical)) is the Cartesian product with componentwise operations. More generally, given a normal subgroup $N \trianglelefteq G$ and a subgroup $H \leq G$ (Definition~Subgroup and Normal Subgroup) with $G = NH$ and $N \cap H = \{e\}$, we say $G$ is the semidirect product $N \rtimes H$, where $H$ acts on $N$ (Definition~Group Action) by conjugation.

Depends on

Group (Classical)
Subgroup and Normal Subgroup
Group Action

Dependency Graph

flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 n71746aac["Group (Classical)"] n3d67b716["Subgroup and Normal Subgroup"] nf7184fc4["Group Action"] n59ff85d3["Direct and Semidirect Product"]:::current n71746aac --> n59ff85d3 n3d67b716 --> n59ff85d3 nf7184fc4 --> n59ff85d3 click n71746aac "../objects/71746aac.html" "_self" click n3d67b716 "../objects/3d67b716.html" "_self" click nf7184fc4 "../objects/f7184fc4.html" "_self"