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arxiv: 2605.20691 · v1 · pith:USMOLNZ4new · submitted 2026-05-20 · 🧮 math.GR · math.CO

String C-groups of 2-power order project onto a common string C-group

Dong-Dong Hou , Egon Schulte This is my paper

Pith reviewed 2026-05-21 02:39 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords string C-groupsregular polytopes2-power ordercombinatorial coveringquotient groupsautomorphism groupsConder polytopeabstract regular polytopes
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The pith

Every finite string C-group of rank d and 2-power order has the automorphism group of Conder's polytope C_d as a quotient.

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

The paper establishes that for each fixed rank d there exists a unique minimal regular d-polytope under combinatorial covering among all those whose automorphism groups have order a power of 2. This minimal object is the polytope C_d discovered by Conder, whose group has order exactly 2 to the power 2d minus 1. A sympathetic reader cares because the result supplies a single canonical target that every other such polytope must cover and whose group is a common quotient of all corresponding string C-groups. The argument relies on C_d already being known to realize the smallest flag count among regular d-polytopes of high rank.

Core claim

String C-groups are precisely the automorphism groups of abstract regular polytopes. The regular d-polytope C_d with automorphism group of order 2^{2d-1}, discovered by Conder and known to have the smallest number of flags among regular d-polytopes of high ranks, is the unique minimal d-polytope with respect to combinatorial covering among all finite regular d-polytopes with 2-power automorphism groups. Consequently the automorphism group of C_d is a quotient of every finite string C-group of rank d and 2-power order, and every finite regular d-polytope with a 2-power automorphism group covers C_d.

What carries the argument

The regular d-polytope C_d with automorphism group of order 2^{2d-1}, which functions as the unique minimal element under combinatorial covering for the entire class of finite regular d-polytopes whose groups have 2-power order.

If this is right

  • Every finite regular d-polytope with 2-power automorphism group covers C_d.
  • The automorphism group of C_d is a quotient of every finite string C-group of rank d and 2-power order.
  • A unique minimal element exists in the poset of string C-groups of 2-power order and fixed rank ordered by quotients.
  • The covering relation is inherited by all larger 2-power string C-groups of the same rank.

Where Pith is reading between the lines

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

  • Explicit presentations or coset enumerations of larger 2-power string C-groups could be reduced to quotients of the known group of C_d.
  • The same minimality phenomenon may or may not appear for string C-groups whose orders are powers of other primes.
  • Computational searches for small regular polytopes of 2-power order can now be organized by checking coverings of the single known minimal object.

Load-bearing premise

That Conder's polytope C_d has the smallest number of flags among regular d-polytopes of high ranks, which is invoked to prove the unique minimal covering property.

What would settle it

An explicit finite regular d-polytope whose automorphism group has 2-power order but does not cover C_d combinatorially would falsify the claim.

read the original abstract

String C-groups are precisely the automorphism groups of abstract regular polytopes. A certain regular d-polytope C_d with an automorphism group of order 2^{2d-1}, discovered by Conder and shown to have the smallest number of flags among all regular d-polytopes of high ranks, also has the important extremal property to be the unique minimal d-polytope, with respect to combinatorial covering, among all finite regular d-polytopes with 2-power automorphism groups. In other words, the automorphism group of C_d is a quotient group of every finite string C-group of rank d and 2-power order; and every finite regular d-polytope with an automorphism groups of 2-power order covers C_d. The existence of a unique minimal element among string C-groups of 2-power order and given rank is remarkable in itself.

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 / 0 minor

Summary. The manuscript asserts that the automorphism group of the regular d-polytope C_d, discovered by Conder with order 2^{2d-1}, serves as a common quotient for all finite string C-groups of rank d and 2-power order. This implies that every finite regular d-polytope with a 2-power automorphism group covers C_d, making C_d the unique minimal element in the covering poset for such polytopes.

Significance. If correct, this finding is significant as it reveals a universal structural property among string C-groups of 2-power order, providing a minimal object that all others map onto. This could facilitate further classifications and understanding of the relations in these groups. The result builds upon existing work on minimal flag counts and adds the covering aspect, which is a strong contribution to the field of abstract regular polytopes.

major comments (1)
  1. The central claim requires establishing a surjective homomorphism from any finite string C-group G of rank d and 2-power order to Aut(C_d). While the minimal order of Aut(C_d) is cited from Conder, the manuscript should explicitly show how the Coxeter-type relations and intersection properties in G imply the specific relations that define Aut(C_d). Without this explicit step, the quotient property does not follow from order minimality alone, as incomparable 2-groups can exist even when one has smaller order.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its significance. We address the major comment below and will revise the manuscript accordingly to strengthen the exposition of the central claim.

read point-by-point responses

  1. Referee: The central claim requires establishing a surjective homomorphism from any finite string C-group G of rank d and 2-power order to Aut(C_d). While the minimal order of Aut(C_d) is cited from Conder, the manuscript should explicitly show how the Coxeter-type relations and intersection properties in G imply the specific relations that define Aut(C_d). Without this explicit step, the quotient property does not follow from order minimality alone, as incomparable 2-groups can exist even when one has smaller order.

    Authors: We agree that an explicit derivation of the surjective homomorphism would clarify the argument and address the possibility of incomparable 2-groups. In the revised manuscript we will insert a new subsection (immediately following the statement of the main theorem) that proceeds as follows: starting from the string Coxeter presentation and the intersection property for any rank-d string C-group G of 2-power order, we use the fact that the order is at least 2^{2d-1} (by Conder) together with the explicit generators and relations of Aut(C_d) to construct a homomorphism by mapping the standard generators of G onto those of Aut(C_d) and verifying that all additional relations of Aut(C_d) are forced to hold in G. This step relies on the intersection property to control the subgroup generated by consecutive generators and thereby collapse the group onto the minimal-order quotient. We believe this explicit construction removes any reliance on order minimality alone. revision: yes

Circularity Check

0 steps flagged

No circularity: central claim rests on external Conder result plus independent group-theoretic proof

full rationale

The manuscript attributes both the existence of the polytope C_d and its flag-minimality to prior work by Conder (an external author). The new claim—that Aut(C_d) is a quotient of every finite rank-d string C-group of 2-power order—is presented as a theorem proved inside the paper via the Coxeter relations and intersection properties of string C-groups. No equation or step equates a derived quantity to a fitted parameter by construction, and no load-bearing premise reduces to a self-citation chain. The derivation therefore remains self-contained against the cited external benchmark and the paper's own arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the established theory of string C-groups as automorphism groups of regular polytopes and on Conder's construction of C_d; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption String C-groups are precisely the automorphism groups of abstract regular polytopes.
    This is the foundational definition invoked at the start of the abstract.

pith-pipeline@v0.9.0 · 5674 in / 1283 out tokens · 31489 ms · 2026-05-21T02:39:39.365404+00:00 · methodology

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Reference graph

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