Questions tagged [group-presentation]

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

455 questions
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Is every finite group an arithmetic group?

Lubotzky-Phillips-Sarnak(LPS88) constructed Ramanujan graphs of degree $d=p+1$ (for an odd prime $p$) as Cayley graphs of projective linear groups with respect to a carefully chosen set of generators ...
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Find a nontrivial presentation of this group of permutations: the identity with $(1234), (13)(24), (4321), (13), (24), (12)(34),$ and $(14)(23)$.

The Question: Find a nontrivial presentation of the permutation group $G$ consisting of the identity with $(1234), (13)(24), (4321), (13), (24), (12)(34),$ and $(14)(23)$. My Attempt: Under the ...
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A presentation of $C_4$ with one generating symbol $b$ and two defining relators.

This is Exercise 1.1.5(c) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Question: Give a presentation of $C_4$ using one ...
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Cyclically Presented Groups.

I suppose I've answered my own question in writing it but some input would be great. The Question(s) What exactly are cyclically presented groups, what are some examples, and where might I find ...
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Check if an especific equation is a consequence of others (noncommutative scenario)

Let $\Sigma = \{a, b, c, ..., x, y, w\}$ be a set of formal variables, $\Sigma^{-1} = \{a^{-1}, b^{-1}, ..., w^{-1}\}$ be the inverses of those variables, and $B = \{ e_1, e_2, ..., e_n \}$ be a set ...
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Outer automorphism of A4 via automorphism of a free group

The question is about an exercise I found in Robinson's book A Course in the Theory of Groups. It says that, given a presentation $\pi$ from a free group $F$ to a group $G$ and said $R=$ker$\pi$, one ...
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Abstract algebra: finding order or group [closed]

Find the order of a group defined by: $$G=\{u,v\,|\,u^4=v^3=1,\,u\cdot v=u^2\cdot v^2\}.$$ When I first looked at the problem it looked like dihedral group but it's not. I am unable to find ...
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Notation and definition on book Presentations of Groups

On the 33th page of the book Presentations of Groups, when talking about the Nielsen's Method to prove Nielsen–Schreier theorem, the authors consider $F$ as any free group, $H$ as any subgroup of $F$, ...
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How to prove that $\langle x, y \mid xyx^{-1}y^{-1}\rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$

I am a bit new to presentations and free groups, so of course I am stuck on a very easy question. I want to prove that $\langle x, y \mid xyx^{-1}y^{-1}\rangle \cong \mathbb{Z} \times \mathbb{Z}$. ...
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Extending a map to a well defined homomorphism

Given a map $f: S \rightarrow H$, with $S$ some set of generators, when and how can $f$ be extended to a well defined homomorphism with $\phi : G \rightarrow H$ with $G$ a group generated by $S$ with ...
Presentation of wreath product $G=S_3 \wr S_3$ of symmetric groups. What is the isomorphism type of $G/[G,G]$?
I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows; Let $G = S_3 \wr S_3$, the permutational wreath product of two symmetric groups of ...