Chiral polyhedra and finite simple groups

Abstract

Abstract. We prove that every finite non-abelian simple group acts as the automorphism group of a chiral polyhedron, apart from the groups P SL2(q), P SL3(q), P SU3(q) and A7.

On cubic graphical regular representations of finite simple groups

“…and PSU 3 (q) to the list of exceptional groups. More specifically, by Theorem 1.4, [8, Theorem [15,Theorem 1.1], and some known results on alternating groups [11] and classical groups [24], we obtain the following result.…”

“…(Here we do not require S to contain an element of odd prime order.) Combining this with[15, Theorem 1.1]…”

“…Note that, by [5, Proposition 3.5.25 (ii)], there is no such an involutory automorphism outside PGL(V ). By [24,Table 14] for n = 8, 3q 15 for n = 10, 5q 22 for n = 12.…”

“…For G = PSL 4 (2) ∼ = A 8 , (2.4) may fail. However, we can see from[8, Theorem 1.3] and[15, Section 5] that G has a cubic GRR with connection set {z, z −1 , y}, where y is an involution and |z| = 7, though 7 / ∈ ppd(2, 4).…”

Finite simple automorphism groups of edge-transitive maps

“…In [16] Leemans and Liebeck proved these results in one direction, showing that all non-abelian finite simple groups except those in L have such a generating pair x, y, and are therefore automorphism groups of orientably regular chiral maps, that is, members of G(2 P ex). For the converse, it is easy to show that L 2 (q) and A 7 have no such generating pairs, and in [2] d'Azevedo Breda and Catalano proved this for the groups L 3 (q) and U 3 (q), thus completing the proof of Theorem 1.3.…”