On construction of certain Fischer embedded subgroups generated by 3-transpositions in Fi22

In this paper, we investigate the Fischer group F-i(22). This group is generated by a conjugacy class of involutions, any non-commuting pair of which has product of order 3. Such involutions are called transpositions and their conjugacy class is denoted by D. Subgroups generated by elements of D are called D-groups as they have been called by Enright [G. M. Enright, The structure and subgroups of the Fischer groups F-22 and F-23, Ph.D. thesis, University of Cambridge (1976)], Fischer embedded or 3-transposition groups. Here, we obtain the following main results: The rank of the groups F - 3.F-i22, the Weyl group W of type E-6 over fields of characteristic 2, the group M of shape 3(6) (sic)U-4(2).2 and the group H-1 = 3 Omega(7)(3) are computed. If G is a 3-transposition group (Fischer embedded) for a class D of transpositions, then rank(G) is defined as max{vertical bar X vertical bar vertical bar X subset of D, all elements in X commute}. We prove that the subgroups Omega(+)(8) (2) . S-3, 3(5) . S-6, 3 . U-4(3) center dot 2 and SU6 (2) are Fischer embedded and we give an explicit construction for them. It is remarkable to mention that Enright studied certain D-groups under a very strong condition that is O-2(G) <= Z(G) >= O-3(G) and G' = G '' where G is the group generated by a proper subset E of D, G' is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.