# Tagged Questions

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

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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 ...
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### Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an ...
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### Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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### For every group $G$ there is a $2$-dimensional cell complex $X_G$ with $\pi_1(X_G)\cong G$.

I am reading Allen Hatcher's Algebraic Topology, and am trying to understand the proof to corollary 1.28: For every group $G$ there is a $2$-dimensional cell complex $X_G$ with $\pi_1(X_G)\cong G$....
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### Presentation of two Groups

Assume that a group $G$ has a presentation $\langle X \mid R \rangle$ and a group $H$ has a presentation $\langle X \mid S \rangle$. If $R \subseteq S$, the $H$ is isomorphic to a quotient of $G$. In ...
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### Universal cover of a CW complex corresponding to an identification space

I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the ...
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### Expository reference on group presentations

I'm looking for expository papers, small books or chapters on the topic of group presentations. I have familiarity with basic abstract algebra (groups, rings, modules, some finite field theory from ...
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### presentation of the symmetric group via transpositions fixing one element

Consider the symmetric group $S_n$. If we use the most popular set of generators $\sigma_1, \sigma_2,\cdots,\sigma_{n-1}$ with $sigma_i$ being the transposition $(i \, i+1)$, it is well known that ...
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### Say we have a presentation of a finite group. Does adding additional relations to the presentation always decrease the size of the group?

It seems like it should, but I'm not sure how to prove it. EDIT: I'm talking about nontrivial new relations here, i. e. ones that do not follow directly from the old ones.
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### On groups with presentations $\langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle$…

$$\langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r)$$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
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### Abstract finite group vs. finite group

What is the difference between the definitions of abstract finite group and finite group? I have some exposure to the finite group theory. But I came to know about abstract finite group from ...
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### Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_d+A_b+A_d$ (where $d$ is dark, damn...). It's ...
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### $\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$ My question : ...
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### Unique homomorphism from a group defined by generators and relations to $S_3$.

Let $G = \langle a,b; a^2b^{-3} = 1 \rangle$ and consider $(1 2), (123) \in S_3$. I need to show there is a unique homomorphism from $G$ to $S_3$ which sends $a$ to $(12)$ and $b$ to $(123)$. I'm not ...
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### Prove a group has a presentation (revisit)

I was actually asking the same question in here. However, the give answer didn't satisfy me. For the reader's convenience I'll re-write the post as follows: We define the following operation on the ...
Say $G$ is a finite group with presentation $\langle S | R \rangle$ and let $C$ be the commutator subgroup of $G$. Then $\langle S | R \cup \{ sts^{-1}t^{-1} \} \rangle$ is a presentation of $G/C$. ...
### Finite subgroups of $PSL(2,R)$
I know $PSL(2,R)$ is $SL(2,R)/SZ(2,R)$ and it is a simple group, but I do not have single clue how to get on finding its group presentation. How can I find its presentation and also I am looking for ...