What is the definition of a minimal presentation of a group?
$begingroup$
I'm working on a problem on the braid monodromy of complex lines arrangements in $mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere.
Let $G$ be a group and let $langle Smid Rrangle$ be a presentation of $G.$ What does it mean that such presentation is minimal?
Thank you very much for everyone will answer and kind regards.
definition group-presentation combinatorial-group-theory
edited Nov 29 '18 at 22:50
Shaun
9,083113683
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
$endgroup$
add a comment |
$begingroup$
I'm working on a problem on the braid monodromy of complex lines arrangements in $mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere.
Let $G$ be a group and let $langle Smid Rrangle$ be a presentation of $G.$ What does it mean that such presentation is minimal?
Thank you very much for everyone will answer and kind regards.
definition group-presentation combinatorial-group-theory
edited Nov 29 '18 at 22:50
Shaun
9,083113683
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
$endgroup$
1
$begingroup$
In most cases, if the group is finite-generated then minimality if referd to the cardinality of $|S|$. That is a presentation is minimal if $S$ is a minimal set of generators.
$endgroup$
– Ofir Schnabel
Jan 21 '15 at 10:58
1
$begingroup$
And what about the number of relators?
$endgroup$
– esaini582
Jan 21 '15 at 13:21
add a comment |
$begingroup$
I'm working on a problem on the braid monodromy of complex lines arrangements in $mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere.
Let $G$ be a group and let $langle Smid Rrangle$ be a presentation of $G.$ What does it mean that such presentation is minimal?
Thank you very much for everyone will answer and kind regards.
definition group-presentation combinatorial-group-theory
edited Nov 29 '18 at 22:50
Shaun
9,083113683
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
$endgroup$
I'm working on a problem on the braid monodromy of complex lines arrangements in $mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere.
Let $G$ be a group and let $langle Smid Rrangle$ be a presentation of $G.$ What does it mean that such presentation is minimal?
Thank you very much for everyone will answer and kind regards.
definition group-presentation combinatorial-group-theory
definition group-presentation combinatorial-group-theory
edited Nov 29 '18 at 22:50
Shaun
9,083113683
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
edited Nov 29 '18 at 22:50
Shaun
9,083113683
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
edited Nov 29 '18 at 22:50
Shaun
9,083113683
edited Nov 29 '18 at 22:50
Shaun
9,083113683
edited Nov 29 '18 at 22:50
Shaun
9,083113683
9,083113683
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
asked Jan 20 '15 at 16:17
esaini582esaini582
2009
2009
1
$begingroup$
In most cases, if the group is finite-generated then minimality if referd to the cardinality of $|S|$. That is a presentation is minimal if $S$ is a minimal set of generators.
$endgroup$
– Ofir Schnabel
Jan 21 '15 at 10:58
1
$begingroup$
And what about the number of relators?
$endgroup$
– esaini582
Jan 21 '15 at 13:21
add a comment |
1
$begingroup$
In most cases, if the group is finite-generated then minimality if referd to the cardinality of $|S|$. That is a presentation is minimal if $S$ is a minimal set of generators.
$endgroup$
– Ofir Schnabel
Jan 21 '15 at 10:58
1
$begingroup$
And what about the number of relators?
$endgroup$
– esaini582
Jan 21 '15 at 13:21
1
1
$begingroup$
In most cases, if the group is finite-generated then minimality if referd to the cardinality of $|S|$. That is a presentation is minimal if $S$ is a minimal set of generators.
$endgroup$
– Ofir Schnabel
Jan 21 '15 at 10:58
$begingroup$
In most cases, if the group is finite-generated then minimality if referd to the cardinality of $|S|$. That is a presentation is minimal if $S$ is a minimal set of generators.
$endgroup$
– Ofir Schnabel
Jan 21 '15 at 10:58
1
1
$begingroup$
And what about the number of relators?
$endgroup$
– esaini582
Jan 21 '15 at 13:21
$begingroup$
And what about the number of relators?
$endgroup$
– esaini582
Jan 21 '15 at 13:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Quoting this question:
For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with $d(G)$ generators and $r(G)$ relators a minimal presentation.
So, using your notation, both $|S|$ and $|R|$ are the smallest they can be for $langle Smid Rrangle$ to be a presentation of $G$.
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Quoting this question:
For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with $d(G)$ generators and $r(G)$ relators a minimal presentation.
So, using your notation, both $|S|$ and $|R|$ are the smallest they can be for $langle Smid Rrangle$ to be a presentation of $G$.
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
$endgroup$
add a comment |
$begingroup$
Quoting this question:
For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with $d(G)$ generators and $r(G)$ relators a minimal presentation.
So, using your notation, both $|S|$ and $|R|$ are the smallest they can be for $langle Smid Rrangle$ to be a presentation of $G$.
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
$endgroup$
add a comment |
$begingroup$
Quoting this question:
For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with $d(G)$ generators and $r(G)$ relators a minimal presentation.
So, using your notation, both $|S|$ and $|R|$ are the smallest they can be for $langle Smid Rrangle$ to be a presentation of $G$.
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
$endgroup$
Quoting this question:
For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with $d(G)$ generators and $r(G)$ relators a minimal presentation.
So, using your notation, both $|S|$ and $|R|$ are the smallest they can be for $langle Smid Rrangle$ to be a presentation of $G$.
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
answered Nov 29 '18 at 22:48
ShaunShaun
9,083113683
9,083113683
add a comment |
add a comment |
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$begingroup$
In most cases, if the group is finite-generated then minimality if referd to the cardinality of $|S|$. That is a presentation is minimal if $S$ is a minimal set of generators.
$endgroup$
– Ofir Schnabel
Jan 21 '15 at 10:58
1
$begingroup$
And what about the number of relators?
$endgroup$
– esaini582
Jan 21 '15 at 13:21