C-groups of Suzuki type
The non-abelian simple groups are generated by their involutions, and so are likely places to look for abstract regular polytopes (which are certain thin geometries), whose automorphism groups are string C-groups. A C-group \((G,\{\rho_0,\ldots,\rho_{r-1}\})\) has involutory generators \(\rho_i\) which satisfy the intersection property \[ \langle \rho_i \mid i \in I \rangle \cap \langle \rho_j \mid i \in J \rangle = \langle \rho_k \mid k \in I \cap J \rangle \] for all \(I,J \subseteq \{0,\ldots,r-1\}\); the string condition says that \(\rho_i\rho_j = \rho_j\rho_i\) whenever \(|i - j| > 1\) (possibly after reordering of the \(\rho_i\)). A group is nearly simple if it lies between a simple group and its automorphism group. After a discussion of the representation of dihedral groups as C-groups and the structure of Suzuki groups, the authors prove the following. Let \(q = 2^{2e+1} \neq 2\), and let \(\mathrm{Sz}(q) \leq G \leq \mathrm{Aut}(\mathrm{Sz}(q))\) be an almost simple group of Suzuki type. If \((G,\{\rho_0,\ldots,\rho_{r-1}\})\) is a C-group, then \(G = \mathrm{Sz}(q)\) and \(r = 3\). Moreover, there exist at least one such string C-group \(G\) and at least one non-C-group \(G\).
scientific article; zbMATH DE number 6504087
scientific article