Editing Presentation of a group (section)

== 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:

:<math>s_1^{a_1} s_2^{a_2} \cdots s_n^{a_n}</math>

where the ''s<sub>i</sub>'' are elements of S, adjacent ''s<sub>i</sub>'' are distinct, and ''a<sub>i</sub>'' 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]] D<sub>8</sub> 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 D<sub>8</sub> is a product of ''r''{{'}}s and ''f''{{'}}s.

However, we have, for example, {{math|1=''rfr'' = ''f''<sup>−1</sup>}}, {{math|1=''r''<sup>7</sup> = ''r''<sup>−1</sup>}}, etc., so such products are ''not unique'' in D<sub>8</sub>. Each such product equivalence can be expressed as an equality to the identity, such as

:{{math|1=''rfrf'' = 1}}, 
:{{math|1=''r''<sup>8</sup> = 1}}, or
:{{math|1=''f''{{px2}}<sup>2</sup> = 1}}.

Informally, we can consider these products on the left hand side as being elements of the free group {{math|1=''F'' = ⟨''r'', ''f''&nbsp;⟩}}, and let {{math|1=''R'' = ⟨''rfrf'', ''r''<sup>8</sup>, ''f''{{px2}}<sup>2</sup>⟩}}. That is, we let ''R'' be the subgroup generated by the strings ''rfrf'', ''r''<sup>8</sup>, ''f''{{px2}}<sup>2</sup>, each of which is also equivalent to 1 when considered as products in D<sub>8</sub>.

If we then let ''N'' be the subgroup of ''F'' generated by all conjugates ''x''<sup>−1</sup>''Rx'' of ''R'', then it follows by definition that every element of ''N'' is a finite product ''x''<sub>1</sub><sup>−1</sup>''r''<sub>1</sub>''x''<sub>1</sub> ... ''x<sub>m</sub>''<sup>−1</sup>''r<sub>m</sub>'' ''x<sub>m</sub>'' of members of such conjugates. It follows that each element of ''N'', when considered as a product in D<sub>8</sub>, will also evaluate to 1; and thus that ''N'' is a normal subgroup of ''F''. Thus D<sub>8</sub> is isomorphic to the [[quotient group]] {{math|1=''F''/''N''}}. We then say that D<sub>8</sub> has presentation

:<math>\langle r, f \mid r^8 = 1, f^2 = 1, (rf)^2 = 1\rangle.</math>

Here the set of generators is {{math|1=''S'' = {{mset|''r'', ''f''&nbsp;}}}}, and the set of relations is {{math|1=''R'' = {''r'' <sup>8</sup> = 1, ''f'' <sup>2</sup> = 1, (''rf'' )<sup>2</sup> = 1} }}.  We often see ''R'' abbreviated, giving the presentation

:<math>\langle r, f \mid r^8 = f^2 = (rf)^2 = 1\rangle.</math>

An even shorter form drops the equality and identity signs, to list just the set of relators, which is {{math|1= {''r'' <sup>8</sup>, ''f'' <sup>2</sup>, (''rf'' )<sup>2</sup>} }}.  Doing this gives the presentation

:<math>\langle r, f \mid r^8, f^2, (rf)^2 \rangle.</math>

All three presentations are equivalent.