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

Abstract

In this paper, we investigate the Fischer group [Formula: see text]. 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 [Formula: see text] and [Formula: see text], 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 [Formula: see text], the Weyl group W of type [Formula: see text] over fields of characteristic 2, the group M of shape [Formula: see text] and the group [Formula: see text] are computed. If G is a 3-transposition group (Fischer embedded) for a class [Formula: see text] of transpositions, then [Formula: see text] is defined as [Formula: see text], all elements in X commute[Formula: see text]. • We prove that the subgroups [Formula: see text] and [Formula: see text] 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 [Formula: see text] and [Formula: see text] where G is the group generated by a proper subset E of D, [Formula: see text] is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.