Orientably-Regular Maps on Twisted Linear Fractional Groups

Abstract

We present an enumeration of orientably-regular maps with automorphism group isomorphic to the twisted linear fractional group M (q 2 ) for any odd prime power q.

Regular and orientably-regular maps with quasiprimitive automorphism groups on vertices

“…Also, we may assume to be working with a 'canonical' copy of PGL(2, q) within PSL(2, q 2 ) consisting of elements ±u ∈ PSL(2, q 2 ) such that tr(θu) = ±u. Besides the groups PGL(2, q), the almost simple 'twisted linear groups' M (q 2 ) for odd prime powers q ≥ 3 are the 'other' family of sharply 3-transitive groups for which enumeration of (orientably-) regular maps is known [9]. All the maps supported by the groups M (q 2 ) turn out to be orientable, and an inspection of the generating sets in [9] quickly shows that all orientably-regular maps with orientation preserving automorphism group isomorphic to M (q 2 ) satisfy the property that the group acts quasiprimitively on their vertex set.…”

Characters of twisted fractional linear groups

“…To be in position to consider characters of the twisted group M(q 2 ) = G = K/L we will need to determine conjugacy classes of G. This was done in [3] for conjugacy of twisted elements with respect to G, and it turns out that the detailed analysis therein furnishes all one needs to determine the conjugacy within G, a subgroup of G of index two. We sum up the corresponding results in what follows, using notation and machinery of [3].…”

“…To be in position to consider characters of the twisted group M(q 2 ) = G = K/L we will need to determine conjugacy classes of G. This was done in [3] for conjugacy of twisted elements with respect to G, and it turns out that the detailed analysis therein furnishes all one needs to determine the conjugacy within G, a subgroup of G of index two. We sum up the corresponding results in what follows, using notation and machinery of [3]. For any non-zero elements a, b of a field F we let dia(a, b) and off(a, b) be the 2 × 2 matrices of GL(2, F ) with, respectively, the diagonal and off-diagonal entries a, b and with remaining entries equal to zero.…”

“…In [1], enumeration of regular hypermaps of a given type on fractional linear groups was obtained with the help of the Frobenius' character formula [4] for counting tuples of group elements with entries in given conjugacy classes; more applications of this type can be found in [8]. There is, however, no corresponding result for regular hypermaps on twisted fractional linear groups, although enumeration of regular maps (of unspecified type) on these groups can be found in [3].…”

“…To avoid fractional transformations we will work with a representation of M(q 2 ) used in [3]), which also differs from the one of [15] and which will prove more suitable for our purposes. For F = GF(q 2 ), q an odd prime power, we let J = GL(2, F ) ⋊ σ , with σ identified with the additive group C 2 in the obvious way and with multiplication given by (A, r)(B, s) = (AB σ r , r + s), where B σ is obtained from B by applying σ to every entry.…”