String C-groups of 2-power order project onto a common string C-group

Dong-Dong Hou; Egon Schulte

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arXiv · 2605.20691

String C-groups of 2-power order project onto a common string C-group

Dong-Dong Hou·Egon Schulte

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Abstract

String C-groups are precisely the automorphism groups of abstract regular polytopes. A certain regular d-polytope C_d with an automorphism group of order 2^{2d-1}, discovered by Conder and shown to have the smallest number of flags among all regular d-polytopes of high ranks, also has the important extremal property to be the unique minimal d-polytope, with respect to combinatorial covering, among all finite regular d-polytopes with 2-power automorphism groups. In other words, the automorphism group of C_d is a quotient group of every finite string C-group of rank d and 2-power order; and every finite regular d-polytope with an automorphism groups of 2-power order covers C_d. The existence of a unique minimal element among string C-groups of 2-power order and given rank is remarkable in itself.

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Dong-Dong Hou, Egon Schulte. 2026-05-20. String C-groups of 2-power order project onto a common string C-group. https://arxiv.org/abs/2605.20691

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