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Finitely presented groups (henceforth referred to as fp-groups) arise
in many branches of mathematics and computation plays a significant role
in their study. Unfortunately, answering most basic questions about a
given fp-group is often computationally difficult. Over the past
50 years a range of computational tools for investigating fp-groups have
been developed and most of these are installed in Magma. Some are
straightforward to use but others are complicated.
The word problem for fp-groups asks whether there is an algorithm
which given any element of an fp-group can decide whether or not it
is the identity. In 1955, Novikov showed that there are specific
fp-groups for which the word problem is insoluble.
Further results show that other decision problems for fp-groups are
also insoluble. Consequently, this limits what we can compute for
fp-groups. The algorithms (or procedures) that have been developed
can provide some information about many fp-groups but there will be
many more groups for which they fail.
The Handbook documentation describing the fp-group facilities runs to
more than 200 pages. In order to make the material more accessible this
chapter is a terse introduction to the more important tools. Less
important intrinsics are omitted and advanced features of those intrinsics
that are discussed are often also omitted. The hope is that new users
will be able to quickly get enough information from this chapter to be
able to make straightforward application of the more important intrinsics
and then dip into the detailed expositions in chapter FINITELY PRESENTED GROUPS
when they become more proficient.
This chapter is concerned with groups defined by general presentations.
Before starting this chapter the user should read the (very short)
chapter FREE GROUPS on free groups which is assumed knowledge for
this chapter. For groups defined by power-conjugate presentations
the user should consult chapters FINITE SOLUBLE GROUPS and POLYCYCLIC GROUPS.
For users interested in automatic and hyperbolic groups most of the
information concerning them can be found in chapter AUTOMATIC AND HYPERBOLIC GROUPS.
The complete list of intrinsics for fp-groups together with their
advanced features can be found in chapter FINITELY PRESENTED GROUPS.
The intrinsics considered here are designed for doing what is sometimes
referred to as combinatorial group theory. A description of the
fundamental algorithms for finitely presented groups can be found in
Sims [Sim94] or Holt [HEO05]. The facilities
provided for fp-groups in Magma fall into the following categories:-
 - Construction of presentations for fp-groups and their simplification.
- Determination of various properties such as non-trivial, finite/infinite,
perfect, small cancellation, large, automatic and hyperbolic.
- Calculations with subgroups of finite index including the enumeration
of all subgroups having index less a specified bound.
- Construction of particular types of quotient group: abelian quotient,
p-quotient, nilpotent quotient, soluble quotient and simple group
quotients.
The corresponding Magma categories are GrpFP for fp-groups
and GrpFPElt for their elements.
Let X be a finite set of symbols, called generators in this context.
Strings can be formed from X by concatenating the symbols x and
x - 1 for x ∈X.
Defining multiplication ( * ) of strings over X as concatenation and
forcing this multiplication rule to satisfy the group axioms puts a
group structure on strings over X. In this group the identity is
the empty string and the elements are called words. A group of
this type is called the free group of rank n, where n is the
cardinality of X. These groups are discussed in detail in chapter
FREE GROUPS.
A relation is an equality between two words while a relator
consists of a word that equals the identity, that is, it is a relation
where the right hand side is understood to be the identity. If a set R
of non-trivial relations are imposed on F its elements are partitioned
into equivalence classes. This set of equivalence classes form a new group.
If X and R are both finite the group they define is called a
finitely-presented group (abbreviated to fp-group) and is the
subject of this chapter. The pair < X | R > is called
a presentation for G. Note that any free group of finite rank
is also an fp-group where the set R is empty.
The set of relations R can be converted into a set of relators and the
relator words generate a subgroup H of F. If N is the normal
closure of H in F then the group G is the quotient group F/N.
This observation is the basis for the way in which fp-groups are
constructed in Magma. The free group of rank n is constructed by
the intrinsic FreeGroup(n). Having constructed an appropriate
free group F, an fp-group G is constructed by specifying relations on
the generators of F that must be satisfied by the corresponding generators
of G. The details of how this is achieved are discussed in the following
sections.
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