The central product of two cyclic groups of order four along their subgroups of order two is abelian of order eight

Given: The objects and hypotheses in the Example.

For groups $G,H$ with central subgroups $Z_1\le Z(G)$, $Z_2\le Z(H)$ and an isomorphism $\alpha :Z_1\to Z_2$, the central product $G{\circ }_{\alpha }H$ is the quotient of $G\times H$ by $N=\{(z,\alpha (z{)}^{-1}):z\in Z_1\}$ (The central product $G{\circ }_{\alpha }H$ of two groups along an isomorphism of central subgroups).

The subgroup $N=\{(z,\alpha (z{)}^{-1}):z\in Z_1\}$ of $G\times H$ is central, hence normal (The identified subgroup used to form a central product is central, hence normal).

The canonical maps $G\to G{\circ }_{\alpha }H$ and $H\to G{\circ }_{\alpha }H$ are injective homomorphisms whose images commute elementwise, generate $G{\circ }_{\alpha }H$, and meet in the image of $Z_1$ (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

$\mathrm{\mid }G{\circ }_{\alpha }H\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }H\mathrm{\mid }\mathrm{/}\mathrm{\mid }{Z}_1\mathrm{\mid }$, the centre of $G{\circ }_{\alpha }H$ is the image of $Z(G)\times Z(H)$, and its derived subgroup is the image of $G^{\mathrm{\prime }}\times H^{\mathrm{\prime }}$ (Order, centre and derived subgroup of a central product).

If $G=⟨g⟩$ is cyclic, then exactly one of the following applies: (Every cyclic group is isomorphic to $(\mathbb{Z},+)$ or to $(\mathbb{Z}\mathrm{/}n,+)$ for its finite order $n\ge 1$).

$G\times H:=\{(g,h):g\in G,h\in H\}$ (The external direct product $G\times H$ with componentwise multiplication).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).