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This is a Quantitative Study study.

September 24, 2025Open Access

Chiral polytopes of order 2pᵐ

Key Points

The study shows that the automorphism group of a chiral polytope is isomorphic to a specific subgroup structure.
It proves that the conditions for tightness in chiral polytopes depend on properties of the Sylow p-subgroup, particularly its metacyclic nature.
Findings indicate that for certain values of m, all chiral polytopes must be tight, revealing significant constraints on their structure.
The research concludes that for higher orders, there can exist non-tight chiral polytopes, expanding our understanding of their classifications.

Abstract

Let P be a chiral polytope with type \k₁, k₂\ and G=Aut (P). Suppose |G|=2pᵐ, where k₁, k₂ 3 and p is an odd prime. Let P be a Sylow p-subgroup of G. We prove that G P Z₂, d (P) =2, P' 1 (so m 3) and up to duality, \k₁, k₂\=\p^{l₁, 2p^l₂\} for some integral l₁, l₂ 1. Moreover, we show that P is tight (k₁k₂=2pᵐ) if and only if P is metacyclic group. Furthermore, if m=3 or 4, then P must be tight, and if m 5, where either m is odd, or m is even and m p+3, there exists a non-tight chiral polytope P.

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Cite This Study

Kong et al. (Thu,) studied this question.

synapsesocial.com/papers/68d6e0fc8b2b6861e4c3f532 https://doi.org/https://doi.org/10.48550/arxiv.2508.20654

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