Given a set of generators of a group $G$, is there a method to find a presentation for $G$ using those...
$begingroup$
Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want).
Suppose I have a small list of generators ${g_1,ldots,g_n}$ for $G$.
Is there an algorithm to find a presentation for $G$ using the given generators?
More specifically, I have an exact sequence of finitely presented groups $$1rightarrow F_2rightarrow Grightarrow Hrightarrow 1$$
Here $F_2$ is the free group on 2 generators, and I have presentations for both $G$ and $H$. I'd like to find a presentation of $G$ where the generators have the form ${x,y,g_1,ldots,g_k}$, where $x,y$ are the two free generators of $F_2$.
It looks like GAP would allow me to do this for finite groups $G$, though the same function does not appear to work for infinite groups.
algorithms exact-sequence gap group-presentation combinatorial-group-theory
edited Dec 3 '18 at 1:08
Shaun
8,888113681
$endgroup$
migrated from mathoverflow.net Apr 5 '14 at 0:29
This question came from our site for professional mathematicians.
add a comment |
$begingroup$
Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want).
Suppose I have a small list of generators ${g_1,ldots,g_n}$ for $G$.
Is there an algorithm to find a presentation for $G$ using the given generators?
More specifically, I have an exact sequence of finitely presented groups $$1rightarrow F_2rightarrow Grightarrow Hrightarrow 1$$
Here $F_2$ is the free group on 2 generators, and I have presentations for both $G$ and $H$. I'd like to find a presentation of $G$ where the generators have the form ${x,y,g_1,ldots,g_k}$, where $x,y$ are the two free generators of $F_2$.
It looks like GAP would allow me to do this for finite groups $G$, though the same function does not appear to work for infinite groups.
algorithms exact-sequence gap group-presentation combinatorial-group-theory
edited Dec 3 '18 at 1:08
Shaun
8,888113681
$endgroup$
migrated from mathoverflow.net Apr 5 '14 at 0:29
This question came from our site for professional mathematicians.
2
$begingroup$
What else do you know about $G$? To run an algorithm, you need some input.
$endgroup$
– Alex Degtyarev
Apr 4 '14 at 5:10
$begingroup$
I have a presentation. It's not that terrible either - 4 generators, 8 relations.
$endgroup$
– Will
Apr 4 '14 at 5:11
6
$begingroup$
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as Lyndon-Schupp.
$endgroup$
– YCor
Apr 4 '14 at 6:07
2
$begingroup$
In general, if you have a presentation on a (finite) set of generators $X$ and you have another set of generators $Y$, and you can express the generators of $X$ in terms of those of $Y$, and those of $Y$ in terms of $X$, then it is routine to compute a presentation on $Y$. I think you can do this in GAP using Tietze transformations. You should ask for help on the GAP forum.
$endgroup$
– Derek Holt
Apr 4 '14 at 7:10
1
$begingroup$
Start with your known presentation of $G$. Express each of $x$ and $y$ as terms built from those generators. Adjoin to your known presentation the two generators $x$ and $y$, and adjoin to the known relations the two equations that express $x$ and $y$ in terms of the original generators.
$endgroup$
– Andreas Blass
Apr 4 '14 at 14:54
add a comment |
$begingroup$
Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want).
Suppose I have a small list of generators ${g_1,ldots,g_n}$ for $G$.
Is there an algorithm to find a presentation for $G$ using the given generators?
More specifically, I have an exact sequence of finitely presented groups $$1rightarrow F_2rightarrow Grightarrow Hrightarrow 1$$
Here $F_2$ is the free group on 2 generators, and I have presentations for both $G$ and $H$. I'd like to find a presentation of $G$ where the generators have the form ${x,y,g_1,ldots,g_k}$, where $x,y$ are the two free generators of $F_2$.
It looks like GAP would allow me to do this for finite groups $G$, though the same function does not appear to work for infinite groups.
algorithms exact-sequence gap group-presentation combinatorial-group-theory
edited Dec 3 '18 at 1:08
Shaun
8,888113681
$endgroup$
Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want).
Suppose I have a small list of generators ${g_1,ldots,g_n}$ for $G$.
Is there an algorithm to find a presentation for $G$ using the given generators?
More specifically, I have an exact sequence of finitely presented groups $$1rightarrow F_2rightarrow Grightarrow Hrightarrow 1$$
Here $F_2$ is the free group on 2 generators, and I have presentations for both $G$ and $H$. I'd like to find a presentation of $G$ where the generators have the form ${x,y,g_1,ldots,g_k}$, where $x,y$ are the two free generators of $F_2$.
It looks like GAP would allow me to do this for finite groups $G$, though the same function does not appear to work for infinite groups.
algorithms exact-sequence gap group-presentation combinatorial-group-theory
algorithms exact-sequence gap group-presentation combinatorial-group-theory
edited Dec 3 '18 at 1:08
Shaun
8,888113681
edited Dec 3 '18 at 1:08
Shaun
8,888113681
edited Dec 3 '18 at 1:08
Shaun
8,888113681
edited Dec 3 '18 at 1:08
Shaun
8,888113681
edited Dec 3 '18 at 1:08
Shaun
8,888113681
8,888113681
asked Apr 4 '14 at 5:00
Will
migrated from mathoverflow.net Apr 5 '14 at 0:29
This question came from our site for professional mathematicians.
migrated from mathoverflow.net Apr 5 '14 at 0:29
This question came from our site for professional mathematicians.
2
$begingroup$
What else do you know about $G$? To run an algorithm, you need some input.
$endgroup$
– Alex Degtyarev
Apr 4 '14 at 5:10
$begingroup$
I have a presentation. It's not that terrible either - 4 generators, 8 relations.
$endgroup$
– Will
Apr 4 '14 at 5:11
6
$begingroup$
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as Lyndon-Schupp.
$endgroup$
– YCor
Apr 4 '14 at 6:07
2
$begingroup$
In general, if you have a presentation on a (finite) set of generators $X$ and you have another set of generators $Y$, and you can express the generators of $X$ in terms of those of $Y$, and those of $Y$ in terms of $X$, then it is routine to compute a presentation on $Y$. I think you can do this in GAP using Tietze transformations. You should ask for help on the GAP forum.
$endgroup$
– Derek Holt
Apr 4 '14 at 7:10
1
$begingroup$
Start with your known presentation of $G$. Express each of $x$ and $y$ as terms built from those generators. Adjoin to your known presentation the two generators $x$ and $y$, and adjoin to the known relations the two equations that express $x$ and $y$ in terms of the original generators.
$endgroup$
– Andreas Blass
Apr 4 '14 at 14:54
add a comment |
2
$begingroup$
What else do you know about $G$? To run an algorithm, you need some input.
$endgroup$
– Alex Degtyarev
Apr 4 '14 at 5:10
$begingroup$
I have a presentation. It's not that terrible either - 4 generators, 8 relations.
$endgroup$
– Will
Apr 4 '14 at 5:11
6
$begingroup$
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as Lyndon-Schupp.
$endgroup$
– YCor
Apr 4 '14 at 6:07
2
$begingroup$
In general, if you have a presentation on a (finite) set of generators $X$ and you have another set of generators $Y$, and you can express the generators of $X$ in terms of those of $Y$, and those of $Y$ in terms of $X$, then it is routine to compute a presentation on $Y$. I think you can do this in GAP using Tietze transformations. You should ask for help on the GAP forum.
$endgroup$
– Derek Holt
Apr 4 '14 at 7:10
1
$begingroup$
Start with your known presentation of $G$. Express each of $x$ and $y$ as terms built from those generators. Adjoin to your known presentation the two generators $x$ and $y$, and adjoin to the known relations the two equations that express $x$ and $y$ in terms of the original generators.
$endgroup$
– Andreas Blass
Apr 4 '14 at 14:54
2
2
$begingroup$
What else do you know about $G$? To run an algorithm, you need some input.
$endgroup$
– Alex Degtyarev
Apr 4 '14 at 5:10
$begingroup$
What else do you know about $G$? To run an algorithm, you need some input.
$endgroup$
– Alex Degtyarev
Apr 4 '14 at 5:10
$begingroup$
I have a presentation. It's not that terrible either - 4 generators, 8 relations.
$endgroup$
– Will
Apr 4 '14 at 5:11
$begingroup$
I have a presentation. It's not that terrible either - 4 generators, 8 relations.
$endgroup$
– Will
Apr 4 '14 at 5:11
6
6
$begingroup$
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as Lyndon-Schupp.
$endgroup$
– YCor
Apr 4 '14 at 6:07
$begingroup$
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as Lyndon-Schupp.
$endgroup$
– YCor
Apr 4 '14 at 6:07
2
2
$begingroup$
In general, if you have a presentation on a (finite) set of generators $X$ and you have another set of generators $Y$, and you can express the generators of $X$ in terms of those of $Y$, and those of $Y$ in terms of $X$, then it is routine to compute a presentation on $Y$. I think you can do this in GAP using Tietze transformations. You should ask for help on the GAP forum.
$endgroup$
– Derek Holt
Apr 4 '14 at 7:10
$begingroup$
In general, if you have a presentation on a (finite) set of generators $X$ and you have another set of generators $Y$, and you can express the generators of $X$ in terms of those of $Y$, and those of $Y$ in terms of $X$, then it is routine to compute a presentation on $Y$. I think you can do this in GAP using Tietze transformations. You should ask for help on the GAP forum.
$endgroup$
– Derek Holt
Apr 4 '14 at 7:10
1
1
$begingroup$
Start with your known presentation of $G$. Express each of $x$ and $y$ as terms built from those generators. Adjoin to your known presentation the two generators $x$ and $y$, and adjoin to the known relations the two equations that express $x$ and $y$ in terms of the original generators.
$endgroup$
– Andreas Blass
Apr 4 '14 at 14:54
$begingroup$
Start with your known presentation of $G$. Express each of $x$ and $y$ as terms built from those generators. Adjoin to your known presentation the two generators $x$ and $y$, and adjoin to the known relations the two equations that express $x$ and $y$ in terms of the original generators.
$endgroup$
– Andreas Blass
Apr 4 '14 at 14:54
add a comment |
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2
$begingroup$
What else do you know about $G$? To run an algorithm, you need some input.
$endgroup$
– Alex Degtyarev
Apr 4 '14 at 5:10
$begingroup$
I have a presentation. It's not that terrible either - 4 generators, 8 relations.
$endgroup$
– Will
Apr 4 '14 at 5:11
6
$begingroup$
Then the answer is yes, it's just the standard way of changing the presentation when we change the generators, it can be found on standard textbooks such as Lyndon-Schupp.
$endgroup$
– YCor
Apr 4 '14 at 6:07
2
$begingroup$
In general, if you have a presentation on a (finite) set of generators $X$ and you have another set of generators $Y$, and you can express the generators of $X$ in terms of those of $Y$, and those of $Y$ in terms of $X$, then it is routine to compute a presentation on $Y$. I think you can do this in GAP using Tietze transformations. You should ask for help on the GAP forum.
$endgroup$
– Derek Holt
Apr 4 '14 at 7:10
1
$begingroup$
Start with your known presentation of $G$. Express each of $x$ and $y$ as terms built from those generators. Adjoin to your known presentation the two generators $x$ and $y$, and adjoin to the known relations the two equations that express $x$ and $y$ in terms of the original generators.
$endgroup$
– Andreas Blass
Apr 4 '14 at 14:54