Central chief factors of $\mathbb Z_2^+\rtimes\mathbb Z_2^\times$ and failure of normal closure for $\overline Z$-groups

Proof. Normal subgroups form a complete lattice. By the maximal-chain principle, every chain extends to a maximal chain $\mathcal D$. Such a chain contains every supremum and infimum of its subfamilies: each of these bounds is comparable with every member of $\mathcal D$, so maximality puts it in $\mathcal D$. Suprema of nonempty chains of subgroups are their unions. Consequently $\mathcal D$ is a normal system.

If $H\in\mathcal D$ and $H<H^+_{\mathcal D}$, every normal subgroup between these two terms is comparable with every member of $\mathcal D$. It belongs to $\mathcal D$ and must equal an endpoint. Hence $H\lessdot H^+_{\mathcal D}$. If all chief factors are central, every nontrivial jump of $\mathcal D$ is central, and the trivial jumps impose no condition. This gives the required central refinement.

Conversely, suppose that $G\in\overline{Z}$ and $H\lessdot K$. Extend $\{H,K\}$ to a maximal chain and then to a central normal system $\mathcal E$. Every term of $\mathcal E$ above $H$ contains $K$: comparability with $K$ and the chief-factor condition exclude any intermediate term. Since $K\in\mathcal E$, we have $H^+_{\mathcal E}=K$. Centrality gives $[G,K]\le H$. ◻

Proof. The set $I_H$ is an additive subgroup stable under multiplication by units, by (7). In a local ring, either $r$ or $1-r$ is a unit. For $a\in I_H$, the first alternative gives $ra\in I_H$ directly; the second gives $ra=a-(1-r)a\in I_H$. Thus $I_H$ is an ideal.

If $t_a=h t_b$ with $h\in H$ and $b\in I$, comparison of linear parts shows that $h=t_c$ for some $c\in I_H$. Then $a=c+b$. The reverse inclusion in (9) follows by multiplying translations.

The linear-part homomorphism $G\to R^\times$ has abelian image. Hence every commutator $gxg^{-1}x^{-1}$ with $x\in K$ lies in $K\cap T(R)=T(I_K)$. The assumed inclusion puts it in $H$. ◻

Proof of the central-chief-factor assertion in Theorem 1. Let $H\lessdot K$, and put $I=I_K$. If $I\le I_H$, Lemma 4 applies. Suppose therefore that $I\nleq I_H$. Since

the chief-factor condition gives $K=H T(I)$.

Now $H\le H T(\mathfrak mI)\le K$. If $H T(\mathfrak mI)=K$, equation (9) gives

The ideal $I$ is finitely generated because $R$ is Noetherian. Nakayama’s lemma, applied to $I/I_H$, would give $I=I_H$, a contradiction. Therefore

By (4), (7) and (12), every $t_a$ with $a\in I$ centralizes $G$ modulo $H$. Since $K=H T(I)$, the whole factor $K/H$ is central. Lemma 3 now gives $G\in\overline{Z}$. ◻

Proof. Conjugation in $N$ sends $t_n$ to $t_n$ or $t_{-n}$, so both subgroups are normal. Characteristic zero makes $n\mapsto t_n$ injective; in particular $t_1\notin B$.

Let $B<L\le A$, and choose $t_n\in L\setminus B$. Subtracting an integral multiple of $3$ shows that $L$ contains $t_1$ or $t_2$. In the latter case $t_3t_2^{-1}=t_1\in L$. Thus $L$ contains every integral power of $t_1$, and $L=A$. This proves the chief-factor assertion even among all intermediate subgroups.

For $s=d_{-1}\in N$, equation (7) gives

Therefore $A/B$ is not central. ◻

Completion of Theorem 1. The subgroup $N$ is the inverse image of $\{1,-1\}$ under the linear-part homomorphism $G\to R^\times$. Since $R^\times$ is abelian, $N\trianglelefteq G$. Lemmas 5 and 3 give $N\notin\overline{Z}$. ◻

Proof of Corollary 2. The ring $\mathbb Z_2$ is a commutative Noetherian local ring of characteristic zero. Its residue field is $\mathbb F_2$. Reduction sends a unit to a unit of $\mathbb F_2$, necessarily $1$, so (4) holds. Theorem 1 applies. ◻