Simpler Tests for Semisparse Subgroups

Michael I. Hartley

Faculty of Engineering and Computer Science, University of Nottingham Malaysia Campus, Jalan Broga, Semenyih, 43500 Selangor, Malaysia

Michael.Hartley@nottingham.edu.my

Annals of Combinatorics 10 (3) p. 343-352 September, 2006

Abstract:

The main results of this article facilitate the search for quotients of regular abstract polytopes.
A common approach in the study of abstract polytopes is to construct polytopes with specified facets and vertex figures. Any nonregular polytope ,
Q may be constructed as a quotient of a regular polytope P by a (so-called) semisparse subgroup of its automorphism group W (which will be a string C-group).
It becomes important, therefore, to be able to identify whether or not a given subgroup N of a string C-group W is semisparse. This article proves a number of properties of semisparse subgroups.
These properties may be used to test for semisparseness in a way which is computationally more efficient than previous methods.
The methods are used to find an example of a section regular polytope of type {6, 3, 3} whose facets are Klein bottles.

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