Chiral Polytopes and Suzuki Simple Groups

Abstract

For each q ¤ 2 an odd power of 2, we show that the Suzuki simple group S D S z.q/ is the automorphism group of considerably more chiral polyhedra than regular polyhedra. Furthermore, we show that S cannot be the automorphism group of an abstract chiral polytope of rank greater than 4. For each almost simple group G such that S < G Ä Aut.S/, we prove that G is not the automorphism group of an abstract chiral polytope of rank greater than 3, and produce examples of chiral 3-polytopes for each such group G.

Chiral polyhedra and finite simple groups

“…Some results in this vein are known. For instance in [13], it is proved that every almost simple group G with socle Sz(q) is the automorphism group of at least one abstract chiral polyhedron. And in [20], it is shown that the only almost simple groups with socle L 2 (q) that are not automorphism groups of abstract chiral polyhedra are L 2 (q), P GL(2, q), and a group of the form L 2 (9).2.…”

“…x := (1,14,17,21,10,5,2,16,18,12,8) (3,6,19,22,15,9,20,23,4,7,11), y := (1,8,6,10,21,22,19,12,11,7,4,5,3,18,9) (2,23,20,16,13) (14,15,17), and t := xy. The pair x, t satisfies (i)-(iii) of Theorem 1.1.…”

“…x := (1,17,23,21,2,7,3,15,4,20,10,6,16,13,19,22,11,18,5,14,9,8,12), y := (1, 20, 2) (3,7,4,17,21,5,18,24,11,22,19,9,14,23,15) (8,13,16,10,12), and t := xy. The pair x, t satisfies (i)-(iii) of Theorem 1.1.…”

Hexagonal Extensions of Toroidal Maps and Hypermaps

“…Such an atlas existed already in the regular case [28]. These atlases turned out to be very inspiring to find patterns and get classification results (see [26,20,27] for instance).…”

On the rank of Suzuki polytopes: an answer to Hubard and Leemans

“…In [1], Isabel Hubard and Dimitri Leemans have embarked in an extensive analysis on the abstract polytopes admitting an almost simple group with socle a Suzuki group as automorphism group. Except for an answer to [1, Conjecture 1], their analysis is very satisfactory and gives a great insight on such abstract polytopes.…”