[Paper Review] String C-groups from groups of order 2^m and exponent at least 2^(m - 3)

Mark L. Loyola|arXiv (Cornell University)|Jul 6, 2016

Coding theory and cryptography13 references3 citations

TL;DR

This paper classifies string C-groups of order $2^m$ with exponent at least $2^{m-3}$, showing that, aside from cyclic and dihedral groups, only two such groups qualify—both of rank 3 and quotients of the string Coxeter group $[4, 2^{m-3}]$. The classification relies on a complete enumeration of relevant 2-groups and their involution sets, leveraging computational group theory and structural analysis of the intersection condition in string C-groups.

ABSTRACT

This work provides a classification of string C-groups of order 2m and exponent at least 2^(m - 3). Prior to the classification, we complete the list of groups of exponent 2^(m - 3) and rank at least 3. The main result states that, aside from the cyclic and dihedral groups, only two groups of exponent at least 2^(m - 3) are connected string C-groups. Both have rank 3 and are quotients of the string Coxeter group W = [4, 2^(m - 3)].

Motivation & Objective

• To classify all string C-groups of order $2^m$ with exponent at least $2^{m-3}$, extending prior work on 2-groups as automorphism groups of regular polytopes.
• To complete the list of finite 2-groups of exponent $2^{m-3}$ and rank at least 3, which are relevant to the classification.
• To resolve a problem posed by Schulte and Weiss on characterizing 2-groups of order $2^m$ that are string C-groups or automorphism groups of chiral polytopes.
• To provide a structural characterization of string C-groups in the context of abstract regular polytopes, particularly for 2-groups with high exponent.

Proposed method

• Enumerate all finite 2-groups of order $2^m$ with exponent $2^{m-2}$ or $2^{m-3}$ and rank at least 3, using known group-theoretic classifications.
• Compute the full set of involutions for each such group to assess their potential as distinguished generators in string C-groups.
• Apply the string condition and the intersection condition to determine which groups can support a string C-group structure.
• Use computational tools like GAP and Magma to verify group properties and relations, especially orders of products of involutions.
• Analyze the structure of the resulting string C-groups by identifying them as quotients of the string Coxeter group $[4, 2^{m-3}]$, which provides a canonical framework.
• Verify that only two non-cyclic, non-dihedral groups satisfy the string C-group axioms under the given exponent and order constraints.

Experimental results

Research questions

• RQ1 Which 2-groups of order $2^m$ and exponent at least $2^{m-3}$ can be realized as string C-groups?
• RQ2 What is the complete list of finite 2-groups of exponent $2^{m-3}$ and rank at least 3, and how do their involution sets constrain string C-group realization?
• RQ3 Are there any string C-groups of order $2^m$ and exponent $\geq 2^{m-3}$ beyond the cyclic and dihedral groups?
• RQ4 Can all such string C-groups be constructed as quotients of the string Coxeter group $[4, 2^{m-3}]$?
• RQ5 What structural properties distinguish the two exceptional string C-groups from other 2-groups of high exponent?

Key findings

• Only two non-cyclic, non-dihedral string C-groups of order $2^m$ and exponent at least $2^{m-3}$ exist, both of rank 3.
• These two groups are quotients of the string Coxeter group $[4, 2^{m-3}]$, which provides a canonical construction framework.
• The classification confirms that no other 2-groups of exponent $\geq 2^{m-3}$ and order $2^m$ admit a string C-group structure beyond the cyclic and dihedral groups.
• The complete list of 2-groups of exponent $2^{m-3}$ and rank $\geq 3$ has been finalized, enabling the classification.
• The intersection condition and string condition are satisfied only in the two exceptional cases, with all others failing due to nontrivial intersections in subgroup generation.
• The results resolve a key open problem in the classification of 2-groups as automorphism groups of regular polytopes, particularly for orders $2^m$ with high exponent.