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

Author:

Bani-Ata Mashhour1

Affiliation:

1. Department of Mathematics, Public Authority for Applied Education and Training (PAAET), Kuwait

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.

Funder

Public Authority for Applied Education and Training

Publisher

World Scientific Pub Co Pte Ltd

Subject

General Mathematics

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