How to convert one presentation into another? Please explain using a dihedral group as an example.
How can we convert a given presentation of a group $G$ into an another presentation?
Would anyone please explain to me by converting two different presentations of a dihedral group?
Thanks in advance.
group-theory group-presentation dihedral-groups combinatorial-group-theory
edited Nov 30 at 2:00
Shaun
8,339113578
asked Nov 20 '16 at 14:09
Prince Thomas
593210
add a comment |
How can we convert a given presentation of a group $G$ into an another presentation?
Would anyone please explain to me by converting two different presentations of a dihedral group?
Thanks in advance.
group-theory group-presentation dihedral-groups combinatorial-group-theory
edited Nov 30 at 2:00
Shaun
8,339113578
asked Nov 20 '16 at 14:09
Prince Thomas
593210
6
See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32
add a comment |
How can we convert a given presentation of a group $G$ into an another presentation?
Would anyone please explain to me by converting two different presentations of a dihedral group?
Thanks in advance.
group-theory group-presentation dihedral-groups combinatorial-group-theory
edited Nov 30 at 2:00
Shaun
8,339113578
asked Nov 20 '16 at 14:09
Prince Thomas
593210
How can we convert a given presentation of a group $G$ into an another presentation?
Would anyone please explain to me by converting two different presentations of a dihedral group?
Thanks in advance.
group-theory group-presentation dihedral-groups combinatorial-group-theory
group-theory group-presentation dihedral-groups combinatorial-group-theory
edited Nov 30 at 2:00
Shaun
8,339113578
asked Nov 20 '16 at 14:09
Prince Thomas
593210
edited Nov 30 at 2:00
Shaun
8,339113578
asked Nov 20 '16 at 14:09
Prince Thomas
593210
edited Nov 30 at 2:00
Shaun
8,339113578
edited Nov 30 at 2:00
Shaun
8,339113578
edited Nov 30 at 2:00
Shaun
8,339113578
8,339113578
asked Nov 20 '16 at 14:09
Prince Thomas
593210
asked Nov 20 '16 at 14:09
Prince Thomas
593210
asked Nov 20 '16 at 14:09
Prince Thomas
593210
593210
6
See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32
add a comment |
6
See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32
6
6
See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32
See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32
add a comment |
1 Answer
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One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.
To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with
$$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$
Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives
$$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$
which is then isomorphic to
$$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$
by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).
Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.
answered Nov 30 at 2:12
Shaun
8,339113578
add a comment |
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One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.
To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with
$$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$
Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives
$$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$
which is then isomorphic to
$$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$
by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).
Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.
answered Nov 30 at 2:12
Shaun
8,339113578
add a comment |
One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.
To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with
$$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$
Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives
$$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$
which is then isomorphic to
$$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$
by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).
Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.
answered Nov 30 at 2:12
Shaun
8,339113578
add a comment |
One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.
To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with
$$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$
Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives
$$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$
which is then isomorphic to
$$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$
by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).
Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.
answered Nov 30 at 2:12
Shaun
8,339113578
One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.
To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with
$$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$
Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives
$$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$
which is then isomorphic to
$$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$
by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).
Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.
answered Nov 30 at 2:12
Shaun
8,339113578
edited Nov 30 at 3:07
answered Nov 30 at 2:12
Shaun
8,339113578
answered Nov 30 at 2:12
Shaun
8,339113578
answered Nov 30 at 2:12
Shaun
8,339113578
8,339113578
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6
See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32