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arxiv: 2605.20658 · v1 · pith:NX64PK77new · submitted 2026-05-20 · 🧮 math.GR · math.CO

Generalized quaternion NCI-groups, NNN-groups and NNND-groups

Jun-Feng Yang , jia-Li Du , Yan-Quan Feng , Young Soo Kwon This is my paper

Pith reviewed 2026-05-21 02:50 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords generalized quaternion groupNCI-groupNNN-groupNNND-groupnormal Cayley graphCayley digraphgroup automorphism
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The pith

The generalized quaternion group Q_{4n} is an NCI-group for every n at least 2, never an NNN-group, and an NNND-group precisely when n is even and at least 6.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper completes the NCI classification for generalized quaternion groups by proving that Q_{4n} has every normal Cayley graph as a CI-graph. It further shows that no such group admits a normal Cayley graph whose automorphism group contains a non-normal regular copy of the group, while certain even orders allow this for digraphs. A reader would care because these results fix exactly how much extra symmetry normal Cayley objects on Q_{4n} can have, finishing a case left open after the corresponding digraph classification. The proofs use the known NDCI behavior of these groups together with their standard automorphism and subgroup structure.

Core claim

We solve the question and prove that Q_{4n} is an NCI-group for every n≥2. Q_{4n} is not an NNN-group for every n≥2, and is an NNND-group if and only if n≥6 and n is even.

What carries the argument

The NCI, NNN and NNND properties of normal Cayley graphs and digraphs on the generalized quaternion group Q_{4n}, determined by whether isomorphisms of such objects come from group automorphisms and whether their full automorphism groups contain extra non-normal regular copies of Q_{4n}.

If this is right

  • Every pair of isomorphic normal Cayley graphs on Q_{4n} must have connection sets that differ by a group automorphism.
  • No normal Cayley graph on Q_{4n} has an automorphism group that properly contains a non-normal regular copy of Q_{4n}.
  • For even n at least 6, there exist normal Cayley digraphs on Q_{4n} whose automorphism groups do contain such extra regular copies.
  • The NCI property holds uniformly across all orders 4n with n≥2, independent of parity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These uniform NCI and NNN results may simplify isomorphism testing algorithms specifically for Cayley graphs generated from quaternion groups.
  • The sharp even-order threshold for the NNND property invites checking whether similar thresholds appear for other families of 2-groups.
  • Knowing the exact cases where extra regular subgroups appear could guide constructions of graphs with controlled symmetry in related algebraic settings.

Load-bearing premise

The arguments rest on the already-known NDCI classification for these groups together with basic facts about their automorphism groups and normal subgroups.

What would settle it

Finding one normal Cayley graph on some Q_{4n} (for example n=4) that is isomorphic to a second normal Cayley graph but whose connection sets are not related by any automorphism of Q_{4n}, or exhibiting an NNN graph on any Q_{4n}.

read the original abstract

A Cayley (di)graph $\Cay(G,S)$ of a finite group $G$ is called CI if, for every Cayley (di)graph $\Cay(G,T)$ of $G$, $\Cay(G,S)\cong \Cay(G,T)$ implies that $S^{\sigma}=T$ for some $\sigma\in \Aut(G)$. The group $G$ is called an NDCI-group (resp. NCI-group) if every normal Cayley digraph (resp. graph) of $G$ is CI. It was shown that the generalized quaternion group $\Q_{4n}$ of order $4n$ ($n\geq 2$) is an NDCI-group if and only if either $n=2$ or $n$ is odd, but its NCI-group classification has been left as an open question. In this paper, we solve the question and prove that $\Q_{4n}$ is an NCI-group for every $n\geq 2$. A normal Cayley (di)graph of a group $G$ is called NNN if its automorphism group contains a non-normal regular subgroup isomorphic to $G$, and $G$ is called an NNND-group (resp. NNN-group) if it admits an NNN Cayley digraph (resp. graph). In this paper, we show that $\Q_{4n}$ is not an NNN-group for every $n\geq 2$, and is an NNND-group if and only if $n\geq 6$ and $n$ is even.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript resolves an open question by proving that every generalized quaternion group Q_{4n} (n≥2) is an NCI-group. It further establishes that Q_{4n} is never an NNN-group and is an NNND-group precisely when n≥6 is even. The arguments rely on the known NDCI classification (Q_{4n} is NDCI iff n=2 or n odd), the standard structure Aut(Q_{4n}) ≅ Z_{2n} ⋊ Z_2, and the lattice of normal subgroups.

Significance. If correct, the results complete the NCI classification for this family and supply new information on the existence of non-normal regular subgroups inside Aut of normal Cayley graphs/digraphs. The work is grounded in standard facts about quaternion groups and supplies explicit conditions that are in principle falsifiable by direct computation for small n.

major comments (1)
  1. [Proof of Theorem 3.1 (or equivalent NCI theorem for even n)] The separation between the NCI and NDCI cases for even n is load-bearing for the main NCI claim. The manuscript treats normal Cayley graphs directly via the action on the connection set rather than reducing to the digraph result; an explicit statement of why the NDCI obstruction does not carry over to the undirected case (e.g., via a concrete proposition or lemma) would strengthen the argument.
minor comments (2)
  1. [Introduction / §2] The definitions of NNN and NNND are introduced after the NCI material; a brief forward reference or consolidated preliminary section would improve readability.
  2. [Proof of the NNND classification] For the NNND claim, the boundary cases n=4 (even but <6) and n=5 (odd) should be accompanied by explicit verification that no NNN digraph exists, even if the general argument covers them.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. The single major comment concerns the clarity of the distinction between the NCI and NDCI cases for even n. We address this below and will incorporate the suggested clarification.

read point-by-point responses

  1. Referee: [Proof of Theorem 3.1 (or equivalent NCI theorem for even n)] The separation between the NCI and NDCI cases for even n is load-bearing for the main NCI claim. The manuscript treats normal Cayley graphs directly via the action on the connection set rather than reducing to the digraph result; an explicit statement of why the NDCI obstruction does not carry over to the undirected case (e.g., via a concrete proposition or lemma) would strengthen the argument.

    Authors: We agree that an explicit statement would improve the exposition. In the revised manuscript we will insert a brief remark (or short lemma) immediately before the proof of the even-n NCI result. The remark will note that the NDCI obstruction for even n relies on the existence of automorphisms that preserve a directed connection set S but fail to preserve its inverse closure; when S is required to be inverse-closed (as it must be for an undirected Cayley graph), these automorphisms are excluded from consideration, so the obstruction does not arise. This distinction is already implicit in our direct analysis of the action on the inverse-closed set, but we will state it explicitly as suggested. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central results on NCI, NNN, and NNND properties for Q_{4n} are established through direct examination of Aut(Q_{4n}) ≅ (Z_{2n} ⋊ Z_2) and the lattice of normal subgroups (cyclic of index 2 and unique cyclic of index 4), applying these to the action on connection sets for normal Cayley graphs and digraphs. The cited prior NDCI classification (for n=2 or odd n) serves as background context separating cases but is not invoked as a load-bearing reduction or self-definition for the new NCI/NNN/NNND claims; the derivations remain independent and rely on standard group-theoretic facts without fitted inputs, ansatzes smuggled via citation, or renaming of known results. No step equates a derived quantity to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of finite group theory and the automorphism group structure of generalized quaternion groups, plus one cited prior classification result. No free parameters or new invented entities appear in the abstract.

axioms (1)
  • standard math Standard axioms and basic facts of finite group theory, including the structure of Aut(Q_{4n}) and normal subgroups of Q_{4n}.
    Invoked for all definitions of Cayley graphs, automorphisms, normality, and the prior NDCI result.

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