Questions tagged [group-presentation]

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

458 questions
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How do you prove that a group specified by a presentation is infinite?

The group: $$G = \left\langle x, y \; \left| \; x^2 = y^3 = (xy)^7 = 1\right. \right\rangle$$ is infinite, or so I've been told. How would I go about proving this? (To prove finiteness of a ...
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Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle$$ is equivalent to the trivial group. I have tried all ...
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Why the group $\langle x,y\mid x^2=y^2\rangle$ is not free?

Why is the group $G= \langle x,y\mid x^2=y^2\rangle$ not free? I can't find any reason like an element of finite order or some subgroup of it that is not free etc.
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Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
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Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to ...
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Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.

I'm trying to prove that the following group has infinite order: $$H=\langle a,b\mid a^{3}=b^{3}=(ab)^{3}=1\rangle.$$ Currently I'm checking on some cases using the relations, but my problem is the ...
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Finitely generated group which is not finitely presented [duplicate]

Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented? I was looking for a finitely presented group with a central ...