Normality Of Quartic Cayley Graphs On Regular p-Groups: A CFSG-Free Approach
Pith reviewed 2026-05-10 15:58 UTC · model grok-4.3
The pith
Every quartic Cayley graph of a regular p-group with p not equal to 2 or 5 is normal.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Relying on a CFSG-free approach, every quartic Cayley graph of a regular p-group, p ≠ 2,5, is normal. Along the way it is also proved that for an arbitrary p-group G with a minimum set {a,b} of two generators, in the corresponding Cayley graph Cay(G,{a,a^{-1},b,b^{-1}}) the induced action of vertex stabilizer on the neighbors' set is contained in the dihedral group D8.
What carries the argument
The induced action of the vertex stabilizer on the four neighbors, shown to be a subgroup of the dihedral group D8, which forces the left regular action of the p-group to be normal in the automorphism group.
Load-bearing premise
The group must be a regular p-group with p not equal to 2 or 5, and the four-element connection set must come from a minimal two-generator set.
What would settle it
A regular p-group G for some prime p not equal to 2 or 5 together with a four-element generating set S such that the automorphism group of Cay(G,S) does not contain the left regular action of G as a normal subgroup.
read the original abstract
Relying on the Classification of Finite Simple Groups it was shown by Feng and Xu (Discrete Math., 2005) that every quartic Cayley graph of a regular $p$-group, $p \neq 2,5$, is normal. In this paper a CFSG-free proof of Feng-Xu theorem is given. Along the way it is also proved that for an arbitrary $p$-group $G$ with a minimum set $\{a,b\}$ of two generators, in the corresponding Cayley graph $\mathrm{Cay}(G,\{a,a^{-1},b,b^{-1}\})$ the induced action of vertex stabilizer on the neighbors' set is contained in the dihedral group $D_8$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to give a proof, independent of the Classification of Finite Simple Groups, that every quartic Cayley graph of a regular p-group (p ≠ 2,5) is normal. It also claims to prove that if G is any p-group with minimal generating set {a,b}, then in the Cayley graph Cay(G,{a,a^{-1},b,b^{-1}}) the induced action of the vertex stabilizer on the four neighbors lies inside the dihedral group D_8.
Significance. If the claims hold, the result is significant: it replaces the 2005 Feng-Xu theorem (which used CFSG) with an elementary argument, removing dependence on the classification for this family of graphs. The auxiliary stabilizer claim may be useful for studying automorphism groups of quartic Cayley graphs on p-groups.
minor comments (1)
- The abstract uses the term 'regular p-group' without recalling its definition; a brief reminder of the standard definition (e.g., the p-group satisfies (xy)^p = x^p y^p c^{p choose 2} for commutators c) would help readers.
Simulated Author's Rebuttal
We thank the referee for their summary of the manuscript and for acknowledging its potential significance in providing a CFSG-free alternative to the Feng-Xu theorem. We address the points raised in the report below.
read point-by-point responses
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Referee: The paper claims to give a proof, independent of the Classification of Finite Simple Groups, that every quartic Cayley graph of a regular p-group (p ≠ 2,5) is normal.
Authors: This is an accurate description of the main theorem. The manuscript contains a complete, self-contained proof that relies only on the structure theory of p-groups, the definition of Cayley graphs, and direct analysis of possible automorphisms fixing a vertex. No appeal is made to the classification at any stage. revision: no
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Referee: It also claims to prove that if G is any p-group with minimal generating set {a,b}, then in the Cayley graph Cay(G,{a,a^{-1},b,b^{-1}}) the induced action of the vertex stabilizer on the four neighbors lies inside the dihedral group D_8.
Authors: This auxiliary statement is proved in full generality for every 2-generated p-group (not merely the regular ones appearing in the main theorem). The argument proceeds by considering the possible images of the generators under an automorphism fixing the identity and showing that any permutation of the four neighbors must preserve the dihedral relations. revision: no
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Referee: If the claims hold, the result is significant: it replaces the 2005 Feng-Xu theorem (which used CFSG) with an elementary argument, removing dependence on the classification for this family of graphs. The auxiliary stabilizer claim may be useful for studying automorphism groups of quartic Cayley graphs on p-groups.
Authors: We agree with the referee's evaluation of the significance. The elementary character of the proof and the utility of the stabilizer result for further work on automorphism groups are precisely the motivations for the paper. revision: no
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Referee: REFEREE RECOMMENDATION: uncertain
Authors: We believe the arguments are complete and correct. If the referee has identified any specific step that requires additional justification or clarification, we are prepared to supply it either in a revised manuscript or in a direct reply. revision: no
Circularity Check
No significant circularity; independent CFSG-free proof
full rationale
The supplied abstract announces a new, CFSG-free proof of the Feng-Xu theorem on normality of quartic Cayley graphs on regular p-groups (p≠2,5) together with an auxiliary stabilizer claim. No equations, lemmas, or derivation steps are visible, so no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations can be exhibited. The cited Feng-Xu result is external prior work by different authors; the present paper replaces rather than re-uses that argument. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.
discussion (0)
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