Questions tagged [group-presentation]

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

455 questions
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How do I prove that group determined by generators and relations is trivial? [closed]

I have a group $\langle a,b\mid a^4=b^2=1, ab^2=b^3a, ba^3=a^2b\rangle$ how do I show that it's trivial?
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Free group and isomorphism

suppose $F$ is a free group generated by $x$ and $y$. Prove that $u=x^2, v=y^3$ generates a subgroup of $F$, and it is isomorphic to the free group on $u,v$ Prove that $u=x^2, v=y^2, z = xy$ ...
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Is there an algorithm/method for actually drawing Van Kampen diagrams?

In theory, given a presentation for a group, I understand what a Van Kampen diagram should look like, but I struggle when it comes to the practicalities of drawing it. Is there an algorithm/method ...
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What is this group $G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle$

Consider the group presentation $$G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle.$$ Is this a known group? What is $G$ isomorphic to? Thanks a lot.
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Notational confusion about HNN-extensions: $G=K \ast_{H,t}$.

Let $G=K \ast_{H,t}$ denote an HNN-extension, i.e., $$H \le K \le G, H^t \le K.$$ Is it true that $\{K, t\}$ is a generating system for $G$? In particular, $G/K$ is cyclic?
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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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Presentations of subgroups

I've been looking into finding the presentation of subgroups $H$ of $G$ from a known presentation of the group $G$. I know that usually, removing generators but keeping the set of relations the same ...
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What is the name of this normal subgroup of the free group?

Given the free group $F_S$, we can define a normal subgroup of $F_S$ as follows. For a given element $w \in F_S$ and $s \in S$, define "the count of $s$", $c_s(w)$, as the number of times $s$ appears ...
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Classifying automorphisms using a group presentation.

I'm new to group presentations and after some playing around with the concept, I've tried to find some relatively clear criteria in terms of the defining relations, that would tell us if a map is an ...
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How to show that elements in these groups pairwise commute? [closed]

Let $F_2$ be the free group on $x,y$ and $N$ the normal subgroup generated by $x^2,y^4,xyx^{-1}y^{-1}$. Consider the group $F_2/N$ and two elements $xN$ and $yN$ in it. How do I show that elements in ...
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Finding $m,n,k$ for which $|G|=mn$ and $G$ is nonabelian, where $G$ is given by $\langle x,y\mid x^m=y^n=1,xy=yx^k\rangle$.

$G$ is defined by the relations $x^m=y^n=1,xy=yx^k$. For which $m,n,k$ does this give a nonabelian group order $mn$? I started playing around with this using GAP, starting with the case where $m=13$....
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Terminology: reversing the order of the generators.

Let $H=\langle Y\mid S\rangle$ be a finite group. Then, for all $h\in H$, we can write $$h=y_1^{\eta_1}\dots y_m^{\eta_m}$$ for some finite $m\in\Bbb N$ and finite $\eta_i\in\Bbb Z$, for $y_i\in Y$....
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Automata defined by group presentations.

I apologise if this question is ill-formed or too broad. It's just for fun. I've thrown in the soft question tag for good measure. Let $G=\langle X\mid R\rangle$ be a group. What can be said in ...
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What is the intersection of the vertices of a face of a simplicial complex?

I am currently reading "Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes" By Cora Welsch and I'm a bit stuck with ...
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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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Show that $\langle{x,y\,\vert\,x^{2}y^{-2}}\rangle$ is not isomorphic to $S_{3}$ [closed]

Could someone give me a suggestion to solve this problem? Show that $\langle{x,y\,\vert\,x^{2}y^{-2}}\rangle$ is not isomorphic to $S_{3}$
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Prove that $\langle{a,b\,\vert\,aba^{-1}b^{2}}\rangle$ is not isomorphic to $\mathbb{Z}/3\mathbb{Z}$.

Could anyone give me a suggestion to solve this problem about group presentations? Prove that $\langle{a,b\,\vert\,aba^{-1}b^{2}}\rangle$ is not isomorphic to $\mathbb{Z}/3\mathbb{Z}$
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Show that the class of the word $aba$ in group $\langle{a,b\,\vert\,a^{2}b^{2} }\rangle$ is not the trivial element. [closed]

Could anyone give me a suggestion to solve this problem about group presentations? Show that the class of the word $aba$ in group $\langle{a,b\mid a^{2}b^{2} }\rangle$ is not the trivial element.
Reading Rotman's book on group theory he states that the following is a presentation of the group of quaternions: $$Q' = \langle x,y \mid x^{2}=y^{2}, \, xyx=y \rangle$$ My question is: how does one ...