Questions tagged [group-presentation]

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

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3
votes
1answer
177 views

Does $SL_2(\mathbb{Z}[\sqrt{2}])$ have a finite presentation?

The modular group group $\text{PSL}_2(\mathbb{Z})$ can be written as something that is nearly a free group on two elements: $$ SL_2(\mathbb{Z}) \simeq \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/3\mathbb{Z}...
3
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2answers
160 views

Minimal presentations and (co)homology groups

I wonder whether there exists a link between the number of generators and relations of a presentation for a given group $G$ and the ranks of its (co)homology groups $H_1(G,\mathbb{Z})$ and $H_2(G, \...
3
votes
1answer
109 views

$\mathbb{C}[x,y\,|\,x^m=1,y^n=1]\cong\mathbb{C}[z\,|\,z^{mn}=1]$ as complex algebras?

If $G$ is any finite abelian group and $K$ an algebraically closed field with $|G|\neq 0$ in $K$, then the group algebra $K[G]\cong M_{n_1}(K)\times\cdots\times M_{n_k}(K)$ by Maschke's and Artin-...
3
votes
1answer
690 views

What are some slick ways to prove that a presentation is actually isomorphic to a given group?

Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or ...
3
votes
1answer
60 views

Automorphisms “killing” and group

I ran into the following concept in passing here. Let $G$ be a group and let $\phi$ be an automorphism of $G$. Let $P$ be a presentation for $G$ with $X$ the set of generators in $P$. Form a new ...
3
votes
1answer
75 views

Proof that finite symmetrized relator sets, which are $C'(1/6)$, with equal normal closures are unique

The following statement is made in the Wikipedia article on small cancellation theory without reference or proof. Can anyone either provide a proof or point me to a reference with a proof? The ...
3
votes
1answer
144 views

Presentation of a non-abelian group of order $p^4$ such that ${G}/{\Phi(G)}\cong \Bbb{Z}_p\times \Bbb{Z}_p$

Let $G$ be a finite non-abelian $p$-group of order $p^4$ and $\frac{G}{\Phi(G)}\cong \Bbb{Z}_p\times \Bbb{Z}_p$, where $p$ is a prime. What is the presentation(s) of $G$?(If $G$ exixsts). Thanks ...
3
votes
2answers
202 views

Shortest words in a group with finite presentation

Suppose we're given a group with presentation G=, where both the generating set and the relations are finite. Given a word $w$ in the elements of $X$, I would like to know whether this word is ...
3
votes
1answer
787 views

Presentation of abelian group

How one can find the abelian group which has a presentation $$\langle x,y,z,w\mid6x+8y+10z+14w, 4x+4y+4z+4w\rangle$$ Is there any way indicates the steps to find such a group? Or just by guesswork ...
3
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0answers
44 views

Understand the free group universal property applied to $D_n$

For $n ≥ 3$ and $D_n$ the dihedral group of order $2n$ with présentation $\langle r, s : r^n = s^2 = srsr = 1\rangle$ prove that for all $(a, b) \in (\Bbb Z/n\Bbb Z)^2$, there exists a morphism $f$ ...
3
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0answers
62 views

How to classify the sets $M$ by their structures?

In this post, we construct a set of matrices with the following properties Given $M$ comprised of $n\times n$ matrices, which satisfies $I_n \in M$ and $0_{n} \not\in M$ If $A,B \in M$, ...
3
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0answers
110 views

Show that $\mathbb{Z} = \langle a, b \mid a^{12} = b,\ ab = ba \rangle$ has dead end elements

This exercise is taken from the book "Office Hours with a Geometric Group Theorist" (Office Hour 15, exercise 8): Exercise: Show that the group $\mathbb{Z}$ has dead end elements with respect to the ...
3
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0answers
55 views

Automorphism group of $\mathbb Z_2^3$

I am trying to find $\text{Aut}(\mathbb Z_2^3)$ and express it in terms of familiar groups and the direct and/or semi direct product. Here's what I have so far: I know that the set of generators $A:=\...
3
votes
1answer
49 views

Minimal size of a generating set for presentations of finite groups

Are there any results on the minimal number of generators required to give a presentation of a finite group? More specifically, given a group G, what is the minimal number of generators needed for a ...
3
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0answers
109 views

Sharpness of the constant $1/6$ in Cancellation Theorem.

Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $R = R^{-1}$ of freely and cyclically reduced relators, and let $\Lambda$ be the girth of $\langle \; S \; | \; R \; ...
3
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0answers
223 views

Trefoil Knot Group

I am studying Knot theory and have gone through the Wirtinger Presentation for the Knot Group. However, I come across the different(at least for me) way of finding the Knot Group. Instead of labeling ...
3
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0answers
53 views

Family of groups with specific presentation

Is there a name for the family of groups given by $n$ generators ($g_1, g_2,\ldots g_n$) and the following relations? $$g_ig_jg_i = g_jg_ig_j,~\forall i,j \in \lbrace 1,\ldots n\rbrace,~i\neq j\\ ...
3
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0answers
62 views

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 ...
3
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0answers
80 views

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 ...
3
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0answers
105 views

Presentation of the special linear group $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$

I'm trying to show that $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$. For this, I defined a homomorphism $f$ from $G$ into $SL_2(\mathbb{Z})$ by $$f(a)=\begin{bmatrix}0& -1\\1&...
3
votes
1answer
435 views

Show that the group is trivial. [duplicate]

Show that the following group is identity: $$G=\langle x,y,z \mid xyx^{-1}=y^{2}\, , \, yzy^{-1}=z^{2}\, , \, zxz^{-1}=x^{2} \rangle.$$ This group is its own derived group. So all I get is group ...
3
votes
1answer
169 views

What is the definition of a minimal presentation of a group?

I'm working on a problem on the braid monodromy of complex lines arrangements in $\mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere. Let ...
3
votes
0answers
76 views

Frattini subgroup of a $p$- group of order $p^4$

Let $p$ be an odd prime and $G$ be a finite non-abelian $p$-group of order $p^4$ with the following presentation: $$\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, c^d=cb, b^d=ba, [a,d]=[b,c]=[a,c]=[a,b]=1\...
3
votes
2answers
121 views

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 "...
3
votes
0answers
135 views

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 ...
2
votes
3answers
94 views

Proving $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$

I'm trying to prove that $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$. If they are isomorphic, then their abelianizations $G/[G, G] = \...
2
votes
2answers
164 views

Identifying $\langle a,b\mid a^2b^2\rangle$.

The Question: What group is $G=\langle a,b\mid a^2b^2\rangle$? Thoughts: I found that the presentation maps onto $\langle a, b\mid a^2, b^2, 1\cdot 1\rangle\cong \mathbb{Z}_2\ast\mathbb{Z}_2$, ...
2
votes
2answers
193 views

Unusual presentation of a cyclic group.

Show that the group with presentation $$\langle a,b| aba^{-1}=b^n, b=(ba)^2\rangle$$ is a cyclic group generated by $a$ and determine its order.
2
votes
1answer
55 views

What is an algorithm for determining if a finitely presented group is finite

Suppose I am given a presentation of a group with a finite number of generators and a finite number of relations. Is there an algorithm for determining if the group is finite? Also, if there is such ...
2
votes
2answers
544 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
votes
3answers
106 views

Show that the group $G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$ has order $27$.

This is Exercise 1.2.21 of Magnus et al's book on combinatorial group theory. The Question: Show that the group $$G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$$ has order $27$...
2
votes
1answer
69 views

In $\langle a, b\mid a^2, b^3, (ab)^2\rangle$, why does $ba=ab^2$?

The Question: In $\langle a, b\mid a^2, b^3, (ab)^2\rangle$, why does $ba=ab^2$? My Attempt: Clearly the presentation defines the group $\mathcal S_3$ under the isomorphism given by $\theta: a\...
2
votes
2answers
73 views

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 $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
2
votes
3answers
506 views

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 ...
2
votes
1answer
1k views

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 ...
2
votes
3answers
60 views

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 ...
2
votes
2answers
106 views

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: $21600, 21602, 21604$?...
2
votes
1answer
43 views

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 ...
2
votes
2answers
97 views

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 ...
2
votes
1answer
103 views

What is the presentation of $\mathbb{Z}_n$

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 $\...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 $G$ ...
2
votes
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 (...
2
votes
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 $...
2
votes
1answer
109 views

centre of a group presentation

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 ...
2
votes
2answers
884 views

Generalized Quaternion Group

Let $w = e^{\Large\frac{i\pi}{n}} \in \mathbb{C}.$ Prove that the matrices $X=\left( \begin{array}{cc} w & 0 \\ 0 & \overline{w} \\ \end{array} \right)$ and $Y = \left( \begin{...
2
votes
1answer
40 views

If we add more relations to a presentation will it always form a quotient group?

Specifically, if I have a presentation $\left<G|R\right>$, and I look at the presentation $\left<G|R,R_1\right>$ is always true that $$\left<G|R,R_1\right>\cong\left<G|R\right>...
2
votes
2answers
67 views

Understanding semidirect product by constructing a non-abelian group of order $21$

I just learnt semidirect product, but only know the basic definition, not gaining the true understanding of it. There is an example that asks the reader to construct a nonabelian group of order $21$. ...