Group $G$ whose center $Z(G)$ is cyclic and with $G/Z(G)$ commutative

Solution 1:

This answer is almost complete and I think (read hope) that it's correct up to the point where I get stuck. (I've posted this as an answer because it's too long for a comment and I should go to bed.)

It suffices by the first isomorphism theorem to find a group $H$ and a surjective homomorphism $\theta : G \to H \times H$ for which $\ker \theta = Z(G)$.

By spelling out the definitions it is easy to see that $G/Z(G)$ is commutative if and only if $g^{-1}h^{-1}gh \in Z(G)$ for all $g,h \in G$, which occurs if and only if $[G,G] \le Z(G)$, where $[G,G]$ is the commutator subgroup of $G$.

This implies that the map $G \to [G,G]$ defined by $x \mapsto g^{-1}x^{-1}gx$ is a homomorphism for fixed $g \in G$, since if $y \in G$ then $$\begin{align} (g^{-1}x^{-1}gx)(g^{-1}y^{-1}gy) &= g^{-1}(x^{-1}gxg^{-1})y^{-1}gy \\ &= g^{-1}y^{-1}(x^{-1}gxg^{-1})gy \\ &= g^{-1}y^{-1}x^{-1}gxy \\ &= g^{-1}(xy)^{-1}g(xy) \end{align}$$ Likewise the map $x \mapsto x^{-1}h^{-1}xh$ is a homomorphism for fixed $h \in G$.

Since $Z(G)$ is cyclic, so is $[G,G]$, and hence it is generated by a single element $g^{-1}h^{-1}gh$ for some $g,h \in G$.

Fix $g,h \in G$ such that $[G,G] = \langle g^{-1}h^{-1}gh \rangle$. Let $H = [G,G]$ and define $\theta : G \to [G,G] \times [G,G]$ by $$\theta(x) = (g^{-1}x^{-1}gx, x^{-1}h^{-1}xh)$$ Now

• We know $\theta$ is a homomorphism by above remarks.
• $\theta$ is surjective. Indeed, if $a \in [G, G] \times [G, G]$ then there exist $m,n$ such that $$a = ( (g^{-1}h^{-1}gh)^m, (g^{-1}h^{-1}gh)^n )$$ But $\theta(h) = ( g^{-1}h^{-1}gh, 1 )$ and $\theta(g) = ( 1, g^{-1}h^{-1}gh )$, so $$a = \theta(h)^m\theta(g)^n = \theta(h^mg^n)$$ hence $a \in \mathrm{im}\, \theta$.
• (This is where I get stuck.) We need to show that $\ker \theta = Z(G)$. The $\supseteq$ direction is obvious. For $\subseteq$, suppose $x \in \ker \theta$ and let $y \in G$. We need to show $xy=yx$. Well $x \in \ker \theta$ implies $xg=gx$ and $xh=hx$, so $x$ commutes with $g$ and $h$ (and their inverses). I'm sure a bit more hacking would yield $xy=yx$ but I haven't quite managed to do it. The fact that $x^{-1}y^{-1}xy \in [G,G]$ and hence $x^{-1}y^{-1}xy = (g^{-1}h^{-1}gh)^n$ for some $n$ should be useful.