This chapter describes a set of tools for working with finite almost-simple
groups (AS-groups). In the program for computing with non-soluble finite groups,
the goal is to reduce the solution of many problems concerning a non-soluble
group G to that of solving the same problem for the non-abelian simple
composition factors of G. We are concerned with very specific types of
computation with AS-groups.
The techniques described in this chapter are under development and are very
incomplete in their coverage. The material falls roughly into two main
categories.
- (a)
- Functions which try to identify a particular group S
known to be almost simple with a standard copy T of that AS-group. In
addition, if such an isomorphism is found, it is often desirable to
explicitly construct it so that questions concerning S can be
answered by mapping them into the "standard" group T. Hence the
recognition functions are divided into those which perform non-constructive recognition (they assert the existence of an
isomorphism between S and T) and those that perform constructive recognition (an explicit isomorphism between S and T
is returned).
- (b)
- Functions which allow the user to determine information
about an AS-group. These functions are usually implemented separately
for each family of simple groups. Thus, for each family of simple groups
our goal is to provide machinery for constructing key properties of any
group T in that family in the context of a standard representation of
the group. Using the isomorphism constructed in (a), this information can
then be transferred back to the user's group S. Examples of such
information include, information about element conjugacy, maximal subgroups,
and Sylow p-subgroups.
The functions described in this chapter do not assume that a BSGS-representation
can be constructed available. Thus, the techniques described here apply to
groups possibly having both much larger order and/or much larger dimension
than those that can be handled with the techniques of Chapters
PERMUTATION GROUPS and
MATRIX GROUPS OVER GENERAL RINGS.
V2.28, 13 July 2023