CI-groups for ternary structures

TL;DR

This work advances the Cayley isomorphism problem for ternary relational structures by characterizing CI-groups within a natural nonabelian family: groups of the form $G\cong C\times D$ where $C$ is cyclic and $D$ is either dihedral or dicyclic with $3 mid |D|$. Building on Pálfy-type reductions and a repertoire of permutation-group tools, the authors prove that such $G$ are CI-groups with respect to ternary structures (and hence graphs and digraphs), with precise corollaries depending on the prime divisors of the odd part and congruence conditions mod $4$. A central technical device is a Pálfy-style normal-block structure theorem (via a chain of normal block systems) for regular subgroups of $S_{2^e n}$ with $|R|=2^e n$, which enables reductions to primitive cases and establishes 3-closure implications. The paper further derives explicit CI-characterizations for several families of semidirect products and analyzes inner holomorphs, showing that 3-closure can obstruct ternary CI-status in natural Frobenius-related constructions. Together, these results extend known CI-status for graphs/digraphs and illuminate the role of block-structure and 3-closure in the ternary Cayley isomorphism problem.

Abstract

We explicitly determine all CI-groups with respect to ternary relational structures that have the form $C \times D$, where $C$ is cyclic and $D$ is either a dicyclic group whose order is not divisible by $3$ or a dihedral group. Such groups are also CI-groups with respect to graphs and digraphs.

Theorems & Definitions (53)

• Theorem 1.1: Pálfy, Palfy1987
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.7
• proof