Symmetric polytopes whose automorphism groups are 2-groups

2025/12/17 by Cunningham, Gabriel, Feng, Yan-Quan, Hou, Dong-Dong +1
Materials Science · Mathematics · #20B25 #51M20 #52B15 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Quasicrystal Structures and Properties

paper · doi:10.48550/arxiv.2512.15511

openalex publication_date 2025/12/17 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28

Abstract

The present work investigates regular, semiregular, and chiral polytopes of any rank d≥ 3, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with automorphism groups of 2-power order: for such polytopes with toroidal sections of rank 3, the various sections of rank 3 can be entirely prescribed (possibly with one exception in the semiregular case). It is also shown that having a 2-group as automorphism group is hereditary under taking universal extensions: the universal extension of a given regular, chiral, or alternating semiregular polytope with a finite or infinite 2-group as automorphism group, is a polytope of one rank higher with an infinite 2-group an automorphism group.

Citations

• String C-groups of order 4p^m
• The number of string C-groups of high rank
• Regular 3-polytopes of order 2ⁿp
• Existence of regular 3-hypertopes with 2ⁿ chambers
• Highly symmetric hypertopes
• Chiral extensions of chiral polytopes
• Semiregular Polytopes and Amalgamated C-groups
• Problems on Polytopes, Their Groups, and Realizations
• Groups of type L₂(q) acting on polytopes

Related

• 2-groups behaving as automorphism groups of regular 3-polytopes
• Two infinite families of chiral polytopes of type \4,4,4\ with solvable automorphism groups
• On chiral polytopes having a group PSL(3,q) as automorphism group
• Regular 3-polytopes of type \n,n\
• A family of regular polytopes of order 4p^m with type \p, 2p\