Semidirect Product

category theory semidirect product Definition 1. Let $G$ be a group, $X ⊂ G$, and $g ∈ G$. Set \[gXg^{-1} =\{y ∈ G\mid y = gxg^{-1}\text{ for some }x ∈ X\}.\] We say that $g$ normalizes $X$ if $gXg^{-1} = X$. Let $A \subset G$. We say that $A$ normalizes $X$ if $a$ normalizes $X$ for every $a \in A$. Let $H \leq G$. The normalizer of $X$ in $H$ is \[ N_H(X)=\left\{h \in H \mid h X h^{-1}=X\right\} . \] Proposition 1. Let $G$ be a group, $H \leq G$, and $X \subset G$. Then $N_H(X) \leq H$. Proof. Since $1 \in H$, and $1 x 1^{-1}=x$ for all $x \in X$, we know $1 \in N_H(X)$. Let $h_1, h_2 \in N_H(X)$. Then \[ h_1 h_2 X\left(h_1 h_2\right)^{-1}=h_1\left(h_2 X h_2^{-1}\right) h_1^{-1}=h_1 X h_1^{-1}=X, \] so $h_1 h_2 \in N_H(X)$. Let $h \in H$, so that $X=h X h^{-1}$. Multiply both sides by $h^{-1}$ on the left and by $h$ on the right to get \[ h^{-1} X h=h^{-1} h X h^{-1} h=X \] thus $h^{-1} \in N_H(X)$. □ Definition 2. Let $G$ be a group and let $X, Y \subset G$. Set \[ X Y=\{x y \in G \mid x \in X \text { and } y \in Y\}\\\quad X^{-1}=\left\{x^{-1} \in G \mid x \in X\right\} . \] Proposition 2. Let $G$ be a group and let $H, K \leq G$. Then $H K \leq G$ if and only if $H K=K H$. Proof. If $M \leq G$, then $M^{-1}=M$. Thus if $H K \leq G$, then $H K=(H K)^{-1}=$ $K^{-1} H^{-1}=K H$. Suppose $H K=K H$. Let $h_1, h_2 \in H$ and $k_1, k_2 \in K$ so that $h_1 k_1$ and $h_2 k_2$ are arbitrary members of $H K$. Since $H K=K H$, there exists $k_3 \in K$ such that $k_1 h_2=h_2 k_3$. Then $h_1 k_1 h_2 k_2=h_1 h_2 k_3 k_2 \in H K$. Let $h \in H$ and $k \in K$ so that $h k$ is an arbitrary member of $H K$. Then $(h k)^{-1}=k^{-1} h^{-1} \in K H=H K$. Thus $H K \leq G$. □ [...]