Questions tagged [group-presentation]

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

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$G_1$ a subquotient of $G_2$ and $G_2$ a subquotient of $G_1$, is $G_1 \cong G_2$?

Sorry for a presumably noobish group theory question. I would like an example of the following: $G_1,G_2$ are finitely presented groups such that there exists finitely presented groups $C_1,C_2$ ...
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Prove that finite and infinite presentations of Thompson group $F$ are isomorphic.

Let $$G=\langle x_0,x_1,\dots\mid x_jx_i=x_ix_{j+1}\text{ for }i<j\rangle,$$ $$H=\langle a,b\mid [ab^{-1},a^{-1}ba],[ab^{-1},a^{-2}ba^2]\rangle,$$ where $[x,y]$ is commutator. These are both ...
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Isomorphy of simple groups of order 360 : a proof with a presentation

It is well known that all simple groups of order 360 are isomorphic with the alternating group $A_{6}$. Cole's original proof is here on StackExchange : $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9)$ ...
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Show that a group is trivial [duplicate]

Let $G$ be the group generated by $a,b,c$ with relations $ab=b^2a$, $bc=c^2b$, and $ca=a^2c$. Show that $G$ is trivial. A related problem is that for $G_1$ generated by $a,b,c,d$ with similar ...
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$G_n$'s mutually non-isomorphic

This question was answered by @Jim Belk And he defined $G_n$ as follows: $$G_n \;=\; \langle a,b \mid [a^{-1}ba,b] = \cdots = [a^{-n}ba^n,b]=1\rangle$$ My question is: Why $G_n$'s are mutually non-...
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computing the fundamental domain for $\Gamma_0(4) \backslash \mathbb{H}$

how do I compute the fundmental domain for a congruence subgroup of $SL(2, \mathbb{Z})$ This region is important because the theta function $\theta(z) = q^{n^2}$ is invariant under two ...
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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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How to show this presentation of the additive group $(\mathbb{Q},+)$?

The task is: Show that $$\langle (x_n)_{ n \in \mathbb{N}} \mid x_n^n = x_{n-1} \text{ for } 1 < n \in \mathbb{N} \rangle$$ is presentation of additive group $(\mathbb{Q},+)$. Can you explain ...
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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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Show that $S_3$ is presented by $\langle a,b\mid a^3, b^2,ab=ba^2\rangle$

Show that group $S_3$ of the objects $x,y,z$ is presented by $\langle a,b \mid a^3,b^2,ab=ba^2\rangle$ under the mapping $a \to (xyz)$ , $b\to (xz)$ I'm confused to what is to be shown in these type ...
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How to prove that this group has at most order 16

If $$G=\langle a,b:a^8=b^2a^4=ab^{-1}ab=e\rangle,$$ how can I prove that $G$ has order at most $16$? I have played with the relations for a while, but am literally stuck. I know that the order of ...
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A group given with presentation is finite or infinite?

The following was an exercise to check if a group is trivial or finite. The group is given by $$G=\langle x,y : yxy^{-1}=x^2, xyx^{-1}=y^2\rangle.$$ Question: Is $G$ a trivial group? or finite group?...