Questions tagged [group-presentation]

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

455 questions
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Finite presentation of a cyclic group

I'm struggling to understand group presentation. There is a theorem that says, every group $G$ is the image of a suitable free group F (free upon a set $X$), and there must exist a homomorphism $\pi$ ...
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Showing the Heisenberg group admits the presentation $\langle A, B, C\mid AC=CA, BC=CB, ABA^{-1}B^{-1}=C\rangle.$

Consider the subset of $SL_3(\Bbb Z)$ consisting of matrices of the form $$\begin{pmatrix} 1 & a & c\\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$ for $a,b,c\in\Bbb Z$. ...
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Quotient of a finitely presented group by a finitely presented group

Is a quotient of a finitely presentable group $G$ by a subgroup $N$ necessarily finitely presentable? What about if the subgroup $N$ is also finitely presentable?
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The additive group of rationals is not finitely generated

The additive group Z of integers is generator by 1 but additive group of rational numbers is not why?
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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 ...
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Understanding group structure using GAP

My question is regarding how to understand structure of group using GAP. Using StructureDescription(G), GAP gives some information like it is direct product or ...
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Presentation of Dihedral Group

Consider the standard presentation of $D_{2n}$: $\langle r, s : r^n = s^2 =1, rs = sr^{-1}\rangle$. I have seen the latter relation given as $sr = r^{-1}s$ a few times. Is this correct, as well?
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Solvability of group presentations with 2 “almost disjoint” relations

I am interested in a certain type of $2$-relator group presentations arising in algebraic topology which have two relators that only contain a single generator in common. Specifically, suppose I have ...
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presentation of subgroup and quotient group.

If we have a presentation $\langle x_i: R_j\rangle$ of $G$, (1) what does the presentation (in terms of the given generators and the given relations) of a subgroup of $G$ looks like? In particular, ...
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Techniques for finding a complete set of relations for a finitely presented group

I am working on a problem where I want to find a presentation of a particular finite group. I have a particular transitive free action I'm interested in, and using this, I've been able to essentially ...
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How to show $\langle x,y \mid x^3=y^3=(xy)^3=1\rangle$ is presentation of an infinite group? [duplicate]

How to show $\langle x,y \mid x^3=y^3=(xy)^3=1 \rangle$ is presentation of an infinite group? This is a result in textbook, but I do not understand the rationale.
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If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...