# Questions tagged [group-presentation]

For questions concerning groups defined via a presentation by generators and relations.

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### $G=\langle S|R\rangle$ is finitely presented simple, then if $w\neq e$ in $G$, $\langle\langle w\cup R \rangle\rangle = F(S)$

I'm having some trouble seeing $G=\langle S|R\rangle$ is finitely presented and simple, then if the world in free group $F(S)$, $w\neq e$ in $G$, $\langle\langle w\cup R \rangle\rangle = F(S)$ ...
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### If the deficiency of a presentation $P$ is $0$ and $P$ is aspherical, then the deficiency of the group $P$ defines is $0$.

I need a reference for the following theorem: If the deficiency of a presentation $P$ is $0$ and $P$ is aspherical, then the deficiency of the group $P$ defines is also $0$. I think it's by ...
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### Finding presentation of a subgroup in GAP

I have a finitely prsented group $G$ and its subgroup $H$. They aren't stored however as fp groups in GAP. I can quite easy obtain some presentation $pr$ of $G$ in. How can I obtain the presentation ...
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### Higman's group is not trivial.

I'm trying to prove that Higman's group is not trivial. In order to do that, first of all I have to define the following groups: $\langle h_{i},h_{i+1}| h_{i+1}h_{i}h_{i+1}^{-1}=h_{i}^2\rangle$ for ...
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### $\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$ My question : ...
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### Surjective Group Homomorphism From Braid Group Into Symmetric Group

I am reading this article on wikipedia. Here's the relevant excerpt: By forgetting how the strands twist and cross, every braid on $n$ strands determines a permutation on $n$ elements. This ...
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### Tietze transformation

I have a question in which I have to transform $\textbf{I.}$ $\langle a,b,c \mid b^2, (bc)^2\rangle$ to $\textbf{II.}$ $\langle x,y,z\mid y^2, z^2\rangle$ using Tietze transformations. My ...
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### Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
### Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$? [duplicate]
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$? I've just been playing around with the relations but I can't seem to get ...