# Questions tagged [group-presentation]

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

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### Presentation of the shift-replace monoid

Let \begin{align} S(f)(x) &= f(x + 1) \\ R_a(f)(x) &= \begin{cases} a & x = 0 \\ f(x) & \text{otherwise} \end{cases} \end{align} What is the ...
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### Is $\langle\ a,b\ \vert\ aba=bab,\ abab=baba\ \rangle$ a presentation of the free group on a single generator?

Is the following a presentation of the free group generated by a single element? $\langle\ a,b\ \vert\ aba=bab,\ abab=baba\ \rangle.$ My thinking is the following: $abab = baba=b(bab)=b^2ab$ by ...
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### Forgetting a Strand in Braid Groups

Let $B_n$ be the braid group of $n$ strings over the unit disk $D$. Let $$d_i:B_n\to B_{n-1}$$ be the operation which is obtained by forgetting the $i$-th strand, $1\leq i\leq n$. Geometrically this "...
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### presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
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### Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
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### Is a finite index subgroup of a finitely presented subgroup finitely presented?

I do know Schreier's theorem, which states that a finite index subgroup of a finitely generated group is finitely generated. Other than this, I have no reason to suspect a positive answer to my ...
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### Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can't find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the ...
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### Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.

I encountered this problem in Sims' "Computation with Finitely Presented Groups". Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial. The book uses coset enumeration or ...
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### Which of these numbers could be the exact number of elements of order $21$ in a group?

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 4.46. Which of the following numbers could be the exact number of elements of order $21$ in a group: $21600, 21602, 21604$?...
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### How to add relations to make groups trivial

Suppose I have a group $G = <x, y|x^2y^5>$, and I want to add a single relation to make G trivial. Is there any way to do this? In general, if we have a group $G = <x, y|r>$, where r is a ...
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### Automata defined by group presentations.

I apologise if this question is ill-formed or too broad. It's just for fun. I've thrown in the soft question tag for good measure. Let $G=\langle X\mid R\rangle$ be a group. What can be said in ...
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I am given $\langle a,b | ab=ba, b^6=1 \rangle$ and I am supposed to compute the group that has this presentation. After racking my brains for a long time, the only thing I can come up with is $\... 2answers 615 views ### Presentation of the additive group of the rational numbers We know that$\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as $$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$ Then ... 1answer 88 views ### Notation of Burnside's group theory book “The theory of finite groups” According to the classification of finite non-abelian groups of order$p^4$in Burnside's book "The theory of finite groups, 1897, pages 87-88" one of the types of these groups is the following ... 1answer 95 views ### A generating set of a finitely generated group A group is called finitely generated if it has a presentation with finite generators. Edit: My original question was vacuous. Suppose that$G$is a finitely generated group and$\{g_i\}_{i\in I}$is ... 2answers 261 views ###$\langle X|\emptyset\rangle\ncong\langle X|R\rangle$for finitely presented groups (exercise in Massey) Let$:F_n$denote the free group of rank$n$. How can I solve the exercise 7.6.3.(b), page 234, in Massey's Algebraic Topology? I'm guessing there's been made a mistake and (b) actually reads "If$G$... 1answer 46 views ### Certain Isomorphic Representations of the dihedral group$D_{3}$Using the following presentation of the dihedral group$D_{3}$\begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} There is one (... 1answer 124 views ### Prove$G \cong \langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$Let$D^3= D \times D \times D$where$D = D_\infty$where we see$D$as the group generated by$\mathbb{Z}$and element$0^*$of order$2$such that$0^*n0^*=-n$for all$n \in \mathbb{Z}$. Letting$...
having trouble showing that an element belongs to a centre of a group presentation. Let $G = \langle x,y,z\mid x^2=y^3=z^3=xyz\rangle$ I have to show that $a = xyz$ belongs to the centre of $G$. I ...