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Presentation of a group
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Everything about Presentation Of A Group totally explained

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators. We then say G has presentation:langle S mid R angle. Informally, G has the above presentation if it's the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it's isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.
   As a simple example, the cyclic group of order n has the presentation » langle a mid a^n = e angle.

where e is the group identity. This may be written equivalently as » langle a mid a^n angle,

since terms that don't include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that include an equals sign.
   Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.
   A closely related but different concept is that of an absolute presentation of a group.

Background

A free group on a set S is a group where each element can be uniquely described as a finite length product of the form:
» s_1^ angle

Some theorems

Every group G has a presentation. To see this consider the free group <G> on G. Since G clearly generates itself one should be able to obtain it by a quotient of <G>. Indeed, by the universal property of free groups there exists a unique group homomorphism φ : <G> → G which covers the identity map. Let K be the kernel of this homomorphism. Then G clearly has the presentation <G|K>. Note that this presentation is highly inefficient as both G and K are much larger than necessary.
   Every finite group has a finite presentation.
   The negative solution to the word problem for groups states that there's a finite presentation <S|R> for which there's no algorithm which, given two words u, v, decides whether u and v describe the same element in the group.

Free product

If G has presentation <S|R> and H has presentation <T|Q> with S and T being disjoint then the free product G * H has presentation <S,T|R,Q>.

Direct product

If G has presentation <S|R> and H has presentation <T|Q> with S and T being disjoint then the direct product of G and H has presentation <S,T|R,Q, [S,T]>. Here [S,T] means that every element from S commutes with every element from T.

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