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''of a group''
 
''of a group''
  
A specification of a [[Group|group]] by generators and relations among them.
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A specification of a [[group]] by generators and relations among them.
  
 
====Comments====
 
====Comments====
Every group can be presented by means of generators and relations. A presentation is finitely generated, respectively finitely related, if the number of generators, respectively relations, is finite. A finite presentation is one with both a finite number of relations and a finite number of generators. A presentation of the symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744301.png" /> of permutations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744302.png" /> letters is as follows: there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744303.png" /> generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744304.png" />, and the relations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744306.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744308.png" />. If the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p0744309.png" /> are removed, one obtains a presentation of the braid group (cf. [[Braid theory|Braid theory]]).
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Every group can be presented by means of generators and relations. A presentation is finitely generated, respectively finitely related, if the number of generators, respectively relations, is finite. A finite presentation is one with both a finite number of relations and a finite number of generators. A presentation of the [[symmetric group]] $S_n$ of permutations on $n$ letters is as follows: there are $n-1$  generators $\sigma_2,\ldots,\sigma_n$, and the relations are $\sigma_i^2 = e$, $\sigma_i\sigma_j = \sigma_j\sigma_i$ if $|i-j| \ge 2$, $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$. If the relations $\sigma_i^2 = e$ are removed, one obtains a presentation of the [[braid group]] $B_n$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443010.png" /> is presented by generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443012.png" />, and relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443014.png" />, one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443015.png" />. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443016.png" /> is the quotient group of the free group on the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443017.png" /> by the normal subgroup generated by the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074430/p07443018.png" />. For details cf. [[#References|[a1]]], Sect. 1.2. Given a presentation of a group, there are systematic ways for obtaining presentations of subgroups and quotient groups.
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If $G$ is presented by generators $G_i,$, $i \in I$, and relations $R_j$, $j \in J$, one writes $G = \langle G_i | R_j \rangle$. In that case $G$ is the quotient group of the [[free group]] on the generators $G_i,$ by the normal subgroup generated by the relations $R_j$. For details cf. [[#References|[a1]]], Sect. 1.2. Given a presentation of a group, there are systematic ways for obtaining presentations of subgroups and quotient groups.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations of groups in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  W.O.J. Moser,  "Generators and relations for discrete groups" , Springer  (1965)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations of groups in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  W.O.J. Moser,  "Generators and relations for discrete groups" , Springer  (1965)</TD></TR>
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</table>
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Latest revision as of 18:24, 8 September 2017

of a group

A specification of a group by generators and relations among them.

Comments

Every group can be presented by means of generators and relations. A presentation is finitely generated, respectively finitely related, if the number of generators, respectively relations, is finite. A finite presentation is one with both a finite number of relations and a finite number of generators. A presentation of the symmetric group $S_n$ of permutations on $n$ letters is as follows: there are $n-1$ generators $\sigma_2,\ldots,\sigma_n$, and the relations are $\sigma_i^2 = e$, $\sigma_i\sigma_j = \sigma_j\sigma_i$ if $|i-j| \ge 2$, $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$. If the relations $\sigma_i^2 = e$ are removed, one obtains a presentation of the braid group $B_n$.

If $G$ is presented by generators $G_i,$, $i \in I$, and relations $R_j$, $j \in J$, one writes $G = \langle G_i | R_j \rangle$. In that case $G$ is the quotient group of the free group on the generators $G_i,$ by the normal subgroup generated by the relations $R_j$. For details cf. [a1], Sect. 1.2. Given a presentation of a group, there are systematic ways for obtaining presentations of subgroups and quotient groups.

References

[a1] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations of groups in terms of generators and relations" , Wiley (Interscience) (1966)
[a2] H.S.M. Coxeter, W.O.J. Moser, "Generators and relations for discrete groups" , Springer (1965)
How to Cite This Entry:
Presentation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Presentation&oldid=14626