CI-groups for ternary structures

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.

💡 Research Summary

The paper investigates the Cayley Isomorphism (CI) property for groups with respect to ternary relational structures, extending the well‑studied cases of graphs (binary structures) and higher‑arity structures. A CI‑group for a class C of combinatorial objects is one for which any two Cayley objects on the group are isomorphic if and only if they are isomorphic via a group automorphism. The authors focus on groups of the form (G=C\times D) where (C) is cyclic and (D) is either a dicyclic group of order not divisible by 3 or a dihedral group.

The paper begins by recalling Palfy’s complete classification of CI‑groups for 4‑ary relational structures: a group is a CI‑group for all (k\ge4) iff its order is 4 or (\gcd(n,\varphi(n))=1). From this, Corollary 1.2 (citing earlier work) restricts any non‑abelian CI‑group for ternary structures to a direct product (U\times V) where (U) is cyclic of odd order satisfying (\gcd(|U|,\varphi(|U|))=1) and (V) is one of (Q_8), a dihedral group (\mathrm{D}{2m}), or a dicyclic group (\mathrm{Dic}{4m}). Moreover, if (V=Q_8) or (V) is dicyclic and a prime (p\mid |U|) then (p\not\equiv1\pmod4).

The first main result (Corollary 6.2) shows that whenever (V) is dihedral and the conditions of Corollary 1.2 hold, the whole product (C\times V) is a CI‑group for ternary relational structures. This generalises Babai’s earlier result that only dihedral groups of order (2p) (with (p) prime) are CI‑groups for ternary structures. The authors prove this by analysing regular subgroups of the automorphism group of a ternary Cayley structure and showing that any two such regular copies of (G) are conjugate; the key tool is a new structural theorem (Theorem 5.4) which forces the existence of a block system consisting of two blocks of size (|C||V|).

The second main result (Corollary 7.7) treats the case where (V) is dicyclic and (3\nmid |V|). It establishes a precise congruence condition: the product (C\times V) is a CI‑group for ternary structures if and only if every prime divisor of the parameter (m) (where (|V|=4m)) satisfies (p\equiv3\pmod4). The proof relies on earlier theorems (Theorem 2.9 and 2.10) which connect the CI‑property to the existence of a block system of four blocks of size (|C||V|) for binary structures, and of two blocks for ternary structures. The authors also show that the same condition guarantees the CI‑property for graphs and digraphs, thereby extending known results for many non‑abelian groups.

A substantial technical portion of the paper (Section 3) generalises Palfy’s reduction from primitive to imprimitive actions. Using Zsigmondy’s theorem on primitive prime divisors, the authors prove Proposition 3.5: for a primitive permutation group of degree (n=2^{e}m) (with (e\le4) and (m) odd square‑free) that contains a semiregular cyclic subgroup with two orbits or a regular abelian subgroup, either the group is 2‑transitive, belongs to a short list of small exceptional groups, or there exists an odd prime (r\mid n) such that a Sylow‑(r) subgroup has order (r). This result is crucial for handling the imprimitive case in the CI‑analysis.

The paper also defines a family (\mathcal R) of groups (Definition 2.6) that includes all potential non‑abelian CI‑groups for ternary structures, and proves that regular actions of any (R\in\mathcal R) inherit the same family properties on blocks (Proposition 2.7). Lemma 2.8 (from Babai) is used repeatedly to translate the CI‑property into a conjugacy condition for regular subgroups of the automorphism group.

In the concluding sections the authors summarise the implications: every group of the form (C\times D) with (C) cyclic and (D) dihedral or dicyclic (under the stated congruence conditions) is a CI‑group for ternary relational structures, and consequently also for coloured digraphs and ordinary graphs. The remaining open cases are when (V=Q_8) or when the dicyclic factor has order divisible by 3; the authors expect Theorem 5.4 to be instrumental in resolving these in future work. Overall, the paper provides a comprehensive classification of CI‑groups for ternary structures within a broad and natural family of non‑abelian groups, bridging group theory, combinatorial design, and permutation group theory.