Minimal permutation representation of Thompson's simple group

Abstract

Mazurova [i] showed that Thompson's sporadic simple group [2] hasapropersubgroup //of minimal index 143,127,000, isomorphic to an extension of the simple group~# (2) by an automorphism of order three. Moreover she computed the character of the representation o£ Thompson's group on the cosets of /~ , and in particular proved that the corresponding permutation group has rank ii. In this paper, we determine the subdegrees and the double stabilizers of this permutation group. Table 1 gives the values of some of the complex characters of the group /~=~#~).Jon all the conjugacy classes of /~, If C is a conjugacy class of elements of ~/ which is not contained in ~/~ then C ~ denotes the union of the class ~ and the class of the inverses of elements o~ ~.For each character ~i in the table, there exist two further irreducible characters of the same degree, namely %f% and ~i~ , where % is an irreducible linear character of ~. Table 2 lists the maximal subgroups of the group /~. Note that each of these is the normalizer of some prime subgroup of H, This subgroup is described in the column headed "normalizer." Furthermore, we shall use the following facts about the normalizers of some of the prime subgroups ~=~Z:

Subgroups of minimal index in polynomial time

“…where µ(S) can be found by checking the description of minimal faithful permutation representations of finite simple groups, which is well-known (for example, see [6,Table 4] for groups of Lie type and [7,Table 4] for sporadic simple groups). Since all steps can be performed in polynomial time, we obtain the required algorithm.…”

A Moonshine Dialogue in Mathematical Physics

“…Phys: It seems not to be widely known that most sporadic groups may be defined as permutation groups P with two generators [25] -acting on a G-set of cardinality n. These representations can be found explicitly in the 'Atlas of finite group representations' [26]. As at the previous section, a P may be seen as 'dessin d'enfant' D endowed with a natural topological structure with Euler characteristic 2 − 2g = B + W + F − n with B black points, W white points, F faces, n edges and genus g (see [13,15,19] for details).…”

“…Phys: It seems not to be widely known that most sporadic groups may be defined as permutation groups P with two generators [25] -acting on a G-set of cardinality n. These representations can be found explicitly in the 'Atlas of finite group representations' [26].…”

Minimal transitive graphs of finite simple groups

“…If the permutation representation of a group G has rank 2, then diagram (2) implies that the corresponding graph is complete, i.e., each vertex is connected with all of the others by edges. The automorphism group of such a graph is isomorphic to a symmetric group whose degree is equal to the index of the stabilizer of a point in G. Clearly, such graphs are minimal only for corresponding alternating groups.…”