Strong map-symmetry of SL(3,K) and PSL(3,K) for every finite field K

Abstract

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

On chiral polytopes having a group PSL(3,q) as automorphism group

“…All groups PSU(3, q) with q < 11 as well as q = 13 or 16 have no chiral polytopes. Our Theorem 1.3 combined with the results in [1] show that if a group P SU (3, q) is the automorphism group of a chiral polytope, this polytope has either rank four or five.…”

“…for most values of q, the group PSL(2, q) is the automorphism group of a rank four chiral polytope. The groups PSL(3, q) and PSU(3, q) have been shown not to be automorphism groups of chiral polyhedra by Breda and Catalano [1].…”

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.…”

“…If an automorphism of G transposes s 1 and s 2 and fixes s 3 , it inverts x and fixes y, contradicting chirality of the corresponding map. Thus no such automorphism exists, so G ∈ G (2). Then G ∈ G(T ) for each T = 2 α , 3 or 4 α by Lemma 3.4(c), and also for T = 2 α ex or 5 α by Lemma 3.3(b), so G ∈ G(T ) for each T = 1.…”

“…In 2016 Martin Mačaj [17] used GAP to apply this method to a number of finite simple groups G, and found that n(G) = 0 for G = U 4 (3) and U 5 (2), even though these groups did not appear in the published lists of exceptions. His result was verified by Marston Conder and Matan Ziv-Av, using Magma and GAP.…”