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In mathematics, a presentation is one method of specifying a group.A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators.
The free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations.It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation.
A more compact way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators {}, the free group generated by F surjects onto the group G. The kernel of this map is called the subgroup of relations, generated by some subset D.
A free group may be defined from a group presentation consisting of a set of generators with no relations. That is, every element is a product of some sequence of generators and their inverses, but these elements do not obey any equations except those trivially following from gg −1 = 1.
A presentation is in terms of generators and relations; formally speaking the presentation is a pair of a set of named generators, and a set of words in the free group on the generators that are taken to be the relations.
A set of relations defines G if every relation in G follows logically from those in using the axioms for a group. A presentation for G is a pair S ∣ R {\displaystyle \langle S\mid {\mathcal {R}}\rangle } , where S is a generating set for G and R {\displaystyle {\mathcal {R}}} is a defining set of relations.
If G is torsion-free then every subgroup of G either contains a free group of rank 2 or is solvable. If G has nontrivial torsion, then every subgroup of G either contains a free group of rank 2, or is cyclic, or is infinite dihedral. [16] Let G be a one-relator group given by presentation .