Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures

Technical Summary: Cyclic-by-Abelian Counterexamples to the Second and Third Zassenhaus Conjectures

Problem Statement
The paper addresses the validity of the Second and Third Zassenhaus Conjectures ((ZC2) and (ZC3)) within the class of finite cyclic-by-abelian groups.

• (ZC2) asserts that every group basis of the integral group ring $\mathbb{Z}G$ is conjugate to $G$ within the unit group $U(\mathbb{Q}G)$.
• (ZC3) asserts that every finite subgroup of the normalized unit group $V(\mathbb{Z}G)$ is conjugate in $U(\mathbb{Q}G)$ to a subgroup of $G$.

While (ZC1) was recently disproven for metabelian groups by Eisele and Margolis, and (ZC2)/(ZC3) were previously disproven for general metabelian groups by Roggenkamp, Scott, Klingler, and Hertweck, the status of these conjectures for the specific subclass of cyclic-by-abelian groups (where the derived subgroup $G'$ is cyclic) remained open. Margolis and del R´ıo had explicitly asked whether (ZC3) holds for this class. The paper aims to resolve this question by constructing a counterexample.

Methodology
The construction follows and extends the framework established by Hertweck's 2002 counterexample. The methodology relies on a uniform integral gluing argument using pullback diagrams of rings and automorphisms.

1. Group Construction:
   The author defines a family of groups $G_r = (M \times N \times Q) \rtimes W$, where:

   • $M \cong C_5$, $N \cong C_3$, and $Q \cong C_r$ (with $\gcd(r, 30)=1$).
   • $W = C_8 \rtimes (C_2 \times C_2)$ is a specific group of order 32 acting diagonally on the normal subgroup $F = M \times N \times Q$.
   • Unlike Hertweck's original example where $Q \cong C_3$ (making $N \times Q \cong C_3^2$ and the derived subgroup non-cyclic), the choice of $Q \cong C_r$ with $\gcd(r, 30)=1$ ensures the derived subgroup $G'_r \cong C_{60r}$ is cyclic, satisfying the cyclic-by-abelian condition.

2. Automorphism Construction:
   The core of the construction is an augmentation-preserving automorphism $\alpha_r \in \text{Aut}(\mathbb{Z}G_r)$. This is built via a "double-quotient pullback" strategy:

   • Quotients: The ring $\mathbb{Z}G$ is decomposed via pullbacks involving quotients $G_3 = G/M$ and $G_5 = G/N$.
   • Local Automorphisms:
     • On the quotient $H = G/MN \cong Q \rtimes W$, an outer automorphism $\sigma$ is defined (swapping a generator $c$ with $zc$).
     • On the larger quotients $G_3$ and $G_5$, the paper proves that all class-preserving automorphisms are inner (Proposition 3.5).
   • Gluing: The author constructs automorphisms on the integral rings of the quotients ($\mathbb{Z}G_3$ and $\mathbb{Z}G_5$) that agree on the common quotient $\mathbb{Z}H$ modulo specific congruences. This involves:
     • Using cyclotomic units and sign automorphisms ($\varepsilon_u, \varepsilon_v$) derived from elements $m, n$.
     • Constructing a specific integral unit $\nu$ and utilizing a unit $t$ in the cyclotomic quotient to satisfy boundary congruences modulo 3 and 5.
     • Ensuring the "integral gluing" is independent of the order of $Q$ (Proposition 6.3).

3. Obstruction Analysis:
   The constructed automorphism $\alpha_r$ is shown to have no Zassenhaus factorization (i.e., it cannot be written as a central automorphism composed with a group automorphism). This is achieved by demonstrating that if such a factorization existed, it would induce an inner automorphism on the quotient $H$, contradicting the fact that the induced map $\sigma_H$ is outer.

Key Results

• Theorem 1.1: For any $r > 1$ with $\gcd(r, 30)=1$, there exists a finite cyclic-by-abelian group $G_r$ and an automorphism $\alpha_r \in \text{Aut}(\mathbb{Z}G_r)$ such that:
  1. $|G_r| = 480r$ and $G'_r \cong C_{60r}$.
  2. $\alpha_r$ has no Zassenhaus factorization.
  3. The image $Y_r = \alpha_r(G_r)$ is a normalized group basis of $\mathbb{Z}G_r$ that is not conjugate to $G_r$ in $U(\mathbb{Q}G_r)$.
• Corollary 1.2: While $Y_r$ is not simultaneously rationally conjugate to $G_r$, every individual element $y \in Y_r$ is rationally conjugate to some element $g \in G_r$.
• Smallest Example: The smallest member of this family occurs at $r=7$, yielding a group of order 3360 with a derived subgroup $C_{420}$.

Significance and Claims
The paper claims to resolve a specific open problem posed by Margolis and del R´ıo regarding the validity of (ZC2) and (ZC3) for cyclic-by-abelian groups.

• Negative Resolution: It establishes that (ZC2) and (ZC3) fail for the class of finite cyclic-by-abelian groups.
• Structural Distinction: The counterexamples highlight a distinction between pointwise rational conjugacy (which holds for individual torsion units in these groups, as per Caicedo, Margolis, and del R´ıo) and simultaneous rational conjugacy of an entire group basis.
• Uniformity: The construction extends Hertweck's example uniformly. The paper emphasizes that the class-preserving obstruction and the integral gluing mechanism are independent of the order of the auxiliary factor $Q$, provided the coprimality condition is met. This allows for a family of counterexamples rather than a single isolated case.

The work does not propose new applications or future experimental directions but strictly focuses on the theoretical disproof of the conjectures within this specific group class, utilizing the machinery of integral group rings, pullback constructions, and class-preserving automorphisms.