Chiral polytopes of order 2p^m

2025/08/28 by Tingting Kong, Yan‐Quan Feng, Kong, Ting-Ting +7
Mathematics · #20B25 #20D15 #52B10 #52B15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2508.20654

openalex publication_date 2025/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P be a chiral polytope with type \k₁, k₂\ and G=Aut(P). Suppose |G|=2p^m, where k₁, k₂≥ 3 and p is an odd prime. Let P be a Sylow p-subgroup of G. We prove that G ≅ P \rtimes 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=2p^m) 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.

Citations

• String C-groups of order 4p^m
• Four infinite families of chiral 3-polytopes of type \4, 8\ with solvable automorphism groups
• On chiral polytopes having a group PSL(3,q) as automorphism group
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• Groups of order at most 6 000 generated by two elements, one of which is an involution, and related structures
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Related

• Tight Chiral Polytopes
• Projective linear groups as automorphism groups of chiral polytopes
• Symmetric polytopes whose automorphism groups are 2-groups
• On chiral polytopes having a group PSL(3,q) as automorphism group
• Two infinite families of chiral polytopes of type \4,4,4\ with solvable automorphism groups