# Tagged Questions

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### 3-Dimensional analogue of a Dihedral Group

I'm reading about dihedral groups right now, and I'm being asked to find the orders of various groups of rigid motions of 3-dimensional polygons. What I want to know, though, is if there's a nice ...
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### Fundamental group of a Cayley graph

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...
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### Is there a way to show that two groups are isomorphic by visual representation(Cayley diagram)?

I got a question asking me to prove that $V_4$ and $C_2 \times C_2$ are isomorphic. I can do this algebraically. However, I am curious if there is there a way to explain this using the diagram? ...
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### Actions of Finite Groups on Trees

Any action of a finite group on a (non-empty) tree has a global fixed point (in the sense that there is a vertex fixed by all group elements or an edge fixed by all group elements). There is a ...
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### Amalgamated product of groups

This is just a problem solving question. Let $G$ be a finitely generated group such that $G=A*_CB$ where $|A:C|=|B:C|=2$ and $A,B$ are finite. Show that $G$ has a finite index subgroup isomorphic to ...
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### An example of a residually finite group which is not Hopf

trying to think of any residually finite group which is not Hopf. Any help?
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### On which structures does the free group 'naturally' act?

One of the best ways to get a handle on a group is to recognize it as isomorphic to a set of symmetries of some structure. The dihedral group of order $2n$ is easily recognized as the set of ...
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### Presentation of a non-trivial group

I'm having a bit of trouble understanding group presentations. For example, I'm reliably informed that the group $$\langle x, y \mid x^2=y^3 \rangle$$ is not the trivial group, but I don't see ...
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### find a special free subsemigroup

It is well-known theorem that for an elementary amenable group $G$, $G$ has exponential growth rate iff $G$ contains non cyclic free semigroup. Now I am interested in the following questions: Let ...
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### A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
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### Prove an inequality on $l^2$ sequences over $F_2$

Denote $F_2$ the free non-abelian group on two letters $a, b$. Note that any element in $F_2$ is just a word formed by letters from the set $\{a,b,a^{-1},b^{-1}\}$, and the group structure is given ...
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### H0w have group theory and fractal geometry been combined?

Has there been a significant tie made between group theory and fractal geometry? What are some ways that they have been tied together? I've been inspired to ask this question by this image of a free ...
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### Baumslag-Solitar Group $G=\langle a,t \mid tat^{-1}=a^k\rangle\cong\mathbb{Z}[1/k]\rtimes\langle t\rangle$?

Let $G$ be the Baumslag-Solitar group $\langle a,t \mid tat^{-1}=a^k\rangle$ and $$\mathbb{Z}[1/k]:=\left\{\frac{x}{k^n}\mid x\in\mathbb{Z},n\in\mathbb{N}\cup\{0\}\right\}.$$ I'm searching for an ...
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### When does the Commensurator of a subgroup of a group $G$ not equal $G$?

Let $H\leq G$ be two groups. I'm interested in the Commensurator $$\mathrm{comm}_G(H)=\{g\in G : gHg^{-1} \cap H \text{ has finite index in both}\}.$$ Obviously, $\mathrm{comm}_G(H)\leq G$. I read on ...
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### JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
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### $U\leq G$ infinite subgroup, $H\leq G$ subgroup of finite index in $G$ $\Rightarrow$ $U\cap H\neq 1$

I have the following question: If I have an infinite subgroup $U$ of a group $G$ and a subgroup $H$ of finite index in $G$ then how can I show that the intersection of $U\cap H$ is non-trivial??? ...
How can I prove that if $S,S'$ are two different finite generating sets of a group $G$ , then the metric spaces induced by the "word metric" are quasi-isometric? The definition of quasi-isometry is: ...
I have the following question: Let $G=A\underset{C}\star B$ be the free product of two groups $A$ and $B$ with amalgam $C$, such that $C\cap vCv^{-1}=1$ for all reduced words $v$ in G with length ...