Every group has a presentation, and in fact many; a presentation is often the most compact way of describing the structure of the group.

Formally, the group is then specified as being isomorphic to a quotient group of a free group, where the free group is given by *S* and the quotient group is specified by the relations *T*.

Table of contents |

2 Formal definition 3 Examples 4 Some theorems 5 Further Reading |

A free group on a set *S* = {*s _{i}*} is a group where each element can be

*x*_{1}^{a1}*x*_{2}^{a2}...*x*_{n}^{an}

If *G* is any group, and *S* is a generating subset of *G*, then every element of *G* is also of the above form; but in general, these products will not uniquely describe an element of *G*.

For example, the dihedral group *D*_{8} can be generated by a rotation, *r*, of order 4; and a flip, *f*, of order 2; and certainly any element of *D*_{8} is a product of *r* 's and *f* 's.

However, we have, for example, *r f r* = *f*, *r*^{3} = *r*^{ -1}, etc.; so such products are not unique in *D*_{8}. Each such product equivalence can be expressed as an equality to the identity; such as

*r f r f*= 1*r*^{ 4}= 1*f*^{ 2}= 1

If we then let *K* be the subgroup of *F* generated by all conjugates *x*^{ -1} *H x* of *H*, then *K* is a normal subgroup of *F*; and each element of *K*, when considered as a product in *D*_{8}, will also evaluate to 1.

This leads us to consider the quotient group *F*/*K*, and the associated homomorphism *U* | *F* → *F*/*K*, with *K* = Ker(*U*). Then *D*_{8} is isomorphic to *F*/*K*; and we write *D*_{8} has presentation ({r,f}; {r^{4}, f^{2}, (rf)^{2}}).

*x*_{1}^{a1} *x*_{2}^{a2} ... *x*_{n}^{an}

For any natural number *n*, the cyclic group *C*_{n} has presentation ({*x*}; {*x*^{n}}).

If *D*_{2n} is the dihedral group with 2*n* elements, then it has presentation ({*x*,*y*}; {*x*^{n}, *y*^{2}, (*xy*)^{2}}).

The symmetric group *S*_{4} has presentation ({*x*,*y*}; {*x*^{3}, *y*^{4}, (*xy*)^{2}}).

Every group *G* has a presentation, in fact infinitely many.

If *G* has presentation (*S*;*T*) and *S* and *T* are finite, then *G* has **finite presentation**.

Every finite group has a finite presentation.

The negative solution to the word problem for groups states that there is no general algorithm which, for a given presentation (S;T) and two words *u*, *v*, decides whether *u* and *v* describe the same element in the group.

*Group Theory*, W. R. Scott, Dover Publications