Presentation of a group

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. We then say G has presentation

⟨ S ∣ R ⟩ . {\displaystyle \langle S\mid R\rangle .}

Informally, G has the above presentation if it is 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 is 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

⟨ a ∣ a n = 1 ⟩ , {\displaystyle \langle a\mid a^{n}=1\rangle ,}

where 1 is the group identity. This may be written equivalently as

⟨ a ∣ a n ⟩ , {\displaystyle \langle a\mid a^{n}\rangle ,}

thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that do 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 a 1 s 2 a 2 ⋯ s n a n {\displaystyle s_{1}^{a_{1}}s_{2}^{a_{2}}\cdots s_{n}^{a_{n}}}

where the si are elements of S, adjacent si are distinct, and ai are non-zero integers (but n may be zero). In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse. 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 D8 of order sixteen can be generated by a rotation r of order 8 and a flip f of order 2, and certainly any element of D8 is a product of rs and fs. However, we have, for example, rfr = f−1, r7 = r−1, etc., so such products are not unique in D8. Each such product equivalence can be expressed as an equality to the identity, such as

rfrf = 1, r8 = 1, or f‍2 = 1. Informally, we can consider these products on the left hand side as being elements of the free group F = ⟨r, f ⟩, and let R = ⟨rfrf, r8, f‍2⟩. That is, we let R be the subgroup generated by the strings rfrf, r8, f‍2, each of which is also equivalent to 1 when considered as products in D8. If we then let N be the subgroup of F generated by all conjugates x−1Rx of R, then it follows by definition that every element of N is a finite product x1−1r1x1 ... xm−1rm xm of members of such conjugates. It follows that each element of N, when considered as a product in D8, will also evaluate to 1; and thus that N is a normal subgroup of F. Thus D8 is isomorphic to the quotient group F/N. We then say that D8 has presentation

⟨ r , f ∣ r 8 = 1 , f 2 = 1 , ( r f ) 2 = 1 ⟩ . {\displaystyle \langle r,f\mid r^{8}=1,f^{2}=1,(rf)^{2}=1\rangle .}

Here the set of generators is S = {r, f }, and the set of relations is R = {r 8 = 1, f 2 = 1, (rf )2 = 1}. We often see R abbreviated, giving the presentation

⟨ r , f ∣ r 8 = f 2 = ( r f ) 2 = 1 ⟩ . {\displaystyle \langle r,f\mid r^{8}=f^{2}=(rf)^{2}=1\rangle .}

An even shorter form drops the equality and identity signs, to list just the set of relators, which is {r 8, f 2, (rf )2}. Doing this gives the presentation

⟨ r , f ∣ r 8 , f 2 , ( r f ) 2 ⟩ . {\displaystyle \langle r,f\mid r^{8},f^{2},(rf)^{2}\rangle .}

All three presentations are equivalent.

Notation Although the notation ⟨S | R⟩ used in this article for a presentation is now the most common, earlier writers used different variations on the same format. Such notations include the following:

⟨S | R⟩ (S | R) {S; R} ⟨S; R⟩

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