2 Central Product

Suppose $G=HK$ and $N⊲K$. Then

$$G/N=(HN/N)(KN/N),$$

is denoted by $\bar{G}=\bar{H}\bar{K}$. Note that a quotient of an exact factorization is not necessarily exact.

Internal Central Product

Def. A group $G$ is called an internal central product of two of its subgroups $A$ and $B$ if it is generated by them, if $ab=ba$ for any two elements $a\in A$ and $b\in B$ and if the intersection $A\cap B$ lies in the center $\mathrm{Z}(G)$. In particular, for $A\cap B=\{1\}$ the central product turns out to be the direct product $A\times B$.

External Central Product

The external central product of $A$ and $B$ can be defined without assuming in advance that $A$ and $B$ are subgroups of a certain group $C$. More explicitly, see the following definition.

Definition

Let $G=H\times K$. Assume that $C_1⩽Z(H)$, $C_2⩽Z(K)$ and $C_1\cong C_2$. Let $\varphi$ be an isomorphism from $C_1$ to $C_2$. Let $N=⟨(x,x^{\varphi }):x\in C_1⟩⊲G$. Then $\bar{G}=\bar{H}\bar{K}$ s.t. $\bar{H}\cong H$ and $\bar{K}\cong K$. The group $\bar{G}=H\times K/N:=H{\circ }_{N}K$ is called the external central product of $H,K$ w.r.t. $N$.

Example. Let $H=Q_8$ and $K=D_8$. Since $Z(H)=⟨c_1⟩\cong Z_2$ and $Z(K)=⟨c_2⟩\cong Z_2$, take $N=⟨(c_1,c_2)⟩⊲G=H\times K$. Then $\bar{G}=H\circ K$.