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This is a Quantitative Study study.

September 24, 2025Open Access

Chiral polytopes of order 2pᵐ

Puntos clave

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.

Resumen

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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