(Internal) Direct Products

definitiongroup-theory

Definition

Group $(G,\cdot )$, $N,H⊴G$. Assume: $N\cap H=\{e_G\}$ $HN=G$ and $hn=nh$ for all $n\in N,h\in H$. Then we say $N,H$ give an internal direct product decomposition of $G$.

Examples

1. $N=\{e_G\},H=G⊴(G,\cdot )$ $N\cap H=\{e_g\}$ $HN=G\cdot e_G=G$ $⟹\{e_G\}$ and $G$ give internal direct product decomposition of $G$.
2. $G=({\mathbb{Z}}_6,+)$ $N=⟨2⟩=\{0,2,4\}$ $H=⟨3⟩=\{0,3\}$ $N\cap H=\{0\}$ ${}^{\prime }H{N}^{\prime }=H+N=\{0,3,2,5=3+2,4,7=1=3+4\}={\mathbb{Z}}_6=G$ $⟹$ $N$ and $H$ give internal direct product decomposition of ${\mathbb{Z}}_6$.

Non-Examples

1. $G=(\mathbb{Z},+)$ $N=⟨2⟩,H=⟨3⟩$ $H+N=\mathbb{Z}:H+N\le \mathbb{Z}$ The generator of $\mathbb{Z}$, $1$, is in $H+N$, but the $H\cap N\ne 0$.
2. $G=D_3$ $H=⟨\tau ⟩,N=⟨\sigma ⟩$ $H\cap N=\{e\}$ $HN=D_3$ But $H$ is not normal, so $H$ and $N$ do not give internal direct product decomposition.

See

• Note - Internal and External Direct Products