Questions tagged [group-presentation]
For questions concerning groups defined via a presentation by generators and relations. Should probably be used along with the general (group-theory) tag.
877
questions
3
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1
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Is this a valid group structure (of order 12)?
Let
$$
G = \big\langle \, a, b, c \colon a^3 = b^2 = c^2 = (ab)^2 = (bc)^2 = 1, ac = ca \big\rangle. \tag{0}
$$
That is, $G$ is a group that has elements $a$, $b$, and $c$ (though these are not the ...
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6
votes
3
answers
147
views
Prove that $S_4$ is isomorphic to a presentation
I would like to prove that $G=\langle a,b \, | \, a^2,b^4,(ab)^3\rangle \cong S_4$.
I tried to list out all the elements in the group presentation and show that it is isomorphic to $S_4$, but it was ...
- 433
0
votes
0
answers
48
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Reference on group-presentations
I'm looking for an introduction to free abelian groups and their presentation, specifically to understand notation such as:
$G=^{ab}\left\langle a,b,c \text{ } | a + 2b + 3c = −a − 2b − 7c = 0 \right\...
- 149
1
vote
0
answers
72
views
Motivation for presentation of group $G$ in Hungerford’s Abstract Algebra
An immediate consequence of Corollary 9.3 and the First Isomorphism Theorem is that any group $G$ is isomorphic to a quotient group $F/N$, where $G =\langle X\rangle$, $F$ is the free group on $X$ and ...
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1
vote
1
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54
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Compute the automorphism group of $D_{10}$ [duplicate]
This exercise will compute the automorphism group of $D_{10}$. The presentation of the group
is $D_{10 }= <r, s |r^5 = s^2 = 1, s^{−1}rs = r^{−1}>$
(a) (i) Show that under any automorphism $σ : ...
- 355
0
votes
1
answer
70
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Identify the presentation of groups [closed]
I'm trying to classify all groups of order $20$. I get two group presentation and want to identify isomorphism type. I think one of these isomorphic to $D_{10}$ and other isomorphic to $Dic_5$. Can ...
- 377
3
votes
1
answer
117
views
Show that $G = \langle a,b \mid a^2b = ba^3\rangle$ is nonabelian.
I was trying to suppose for a contradiction that it is abelian, then:
$$ba^3 = a^2b = ba^2\implies a = 1$$
Then we have $G = \langle a,b\mid a = 1\rangle$, which I believe is isomorphic to $\mathbb{Z}$...
- 55
9
votes
2
answers
403
views
Show, using presentations, that $\Bbb Z_m\times\Bbb Z_n\cong\Bbb Z_{{\rm lcm}(m,n)}\times \Bbb Z_{\gcd(m,n)} .$
Note: This is an alternative-proof question and thus is not a duplicate.
The Question:
Show, using group presentations, that $$\Bbb Z_m\times\Bbb Z_n\cong\Bbb Z_{{\rm lcm}(m,n)}\times \Bbb Z_{\gcd(m,...
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2
votes
1
answer
58
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Show that every automorphism of $F_{20} = \langle a, b \mid a^5 = b^4 = 1, bab^{-1} = a^3\rangle $ is an inner automorphism
Show that every automorphism of $F_{20} = \langle a, b \mid a^5 = b^4 = 1, bab^{-1} = a^3\rangle $ is an inner automorphism.
I found $Z(F_{20})={1}$ and it follows $F_{20} \cong \text{Inn}(F_{20})$. ...
- 377
12
votes
2
answers
172
views
Is $\Bbb Z^3$ a one-relator group?
I understand that:
$\Bbb Z^0 = \langle a \mid a \rangle$
$\Bbb Z^1 = \langle a, b \mid b \rangle$
$\Bbb Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$
but is it possible for $\Bbb Z^3$ to be ...
- 225
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0
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24
views
What are the elements of the Dihedral Group $D_8$ when given by $D_8 = \langle a, b \mid a^2 = b^2 = (ab)^4=1\rangle?$ [duplicate]
I'm very confused since I've always seen the Dihedral Group $D_8$ given as
$$D_8 = \langle a, b \mid a^4 = b^2 =1, ab= a^3b\rangle.$$
I'm given the Definition
$$D_8 = \langle a, b \mid a^2 = b^2 = (ab)...
- 9
2
votes
0
answers
119
views
When can we say $H$ is a subgroup of $G$ given their presentations?
Consider $F_2=\langle a,b \rangle$, the free group on $2$ letters.
If I look at $H=\langle ab,ba^{-1}, bab^{-1} \rangle$ (the generators are arbitrary), since each generator can be made from $a$ and $...
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1
answer
53
views
Presentation of Klein group: Show $G=\langle a,b\mid a^2,b^2,(ab)^2,(ba)^2,ab^2a\rangle$ is iso to $K_4=\langle x,y\mid x^2,y^2,xyx^{-1}y^{-1}\rangle$
Let
$$G = \langle a, b \mid a^2, b^2, (ab)^2, (ba)^2, a b^2 a \rangle.$$
I need to show that this group is isomorphic to Klein Group
$$K_4 = \langle x,y\mid x^2, y^2, xyx^{-1} y^{-1}\rangle.$$
I prove ...
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0
votes
0
answers
88
views
How to prove that a group affords no epimorphism in $S_3$
Good evening,
I want to prove that the group $G = \langle a, d \mid a d a^{-1} d a = d a d^{-1} a d \rangle$ affords no epimorphism in $S_3$ (the group of the permutations of $\{1, 2, 3\}$). I tried ...
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1
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61
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If $\forall n\in\Bbb N$, if $g\in G$ with $|g|=n$ implies $\forall m\in\Bbb N,\exists h_m\in G,|h_m|=n^m,$ then can non-torsion-free $G$ be f.p.?
Motivation and Details:
Thinking back a couple of years ago, when I did combinatorial group theory a lot, I decided to explore the idea that whenever an element of a group has finite order $n$, then ...
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2
votes
1
answer
81
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Does a presentation of a group correspond to a unique group?
The definition of presentation of a group is as follows:
If $G$ is generated by a subset $S$ and there is some collection of relations, say $R_{1}, R_{2}, \dots, R_{m}$(here each $R_{i}$ is an ...
- 123
4
votes
1
answer
78
views
Proof of the Wirtinger Presentation using Van Kampen Theorem
I have some difficulties understanding a proof of the Wirtinger presentation using the Van Kampen theorem, found in John Stiwell's "Classical Topology and Combinatorial Group Theory".
I ...
- 175
1
vote
1
answer
61
views
What is the free product with amalgamation with the trivial group?
I am working on algebraic topology. I am trying to prove the Wirtinger presentation using the Van Kampen theorem.
However, I have some difficulties understanding the concept of free product with ...
- 175
2
votes
0
answers
77
views
HNN-extension and Centralizer
I am currently studying the book of Graham Higman and Elizabeth Scott, The Existentially Closed Groups, London Mathematical Society Monographs New Series, Clarendon Press Oxford, 1988. In the Section ...
- 21
3
votes
0
answers
56
views
Equivalence of two presentations for a quotient of the braid group
I'm considering two presentations for quotients of the 3-strand braid group $B_3$ that I believe should be equivalent (i.e. yield the same quotient). There is an integer parameter $d \ge 7$ involved, ...
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1
vote
0
answers
47
views
Normal subgroup of fundamental group of Klein Bottle
Let $K$ the Klein Bottle and $\pi_1(K) = \langle a,b \mid b a b a^{-1} \rangle $ be the fundamental group of the Klein bottle. Observe that $\langle b \rangle $ is a normal subgroup of $\pi_1(K)$, ...
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3
votes
1
answer
134
views
Determining whether two groups are isomorphic based on Wirtinger presentation
I am working on knot theory and basic algebraic topology. In order to prove that the figure eight knot and the trefoil knot are not isotopic, I have to show that their knot groups (i.e. the ...
- 175
1
vote
1
answer
82
views
Writing down a finite axiomatization of a group given by finite presentation
I am working on a problem and I am wondering if what I am trying can even be done.
Assume that we are given a finite group $G$ by it's presentation via generators and relations. For example $$G = \...
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1
vote
1
answer
68
views
Are the only finite orders of elements of this group in $\{p^k\mid k\in\Bbb N\cup\{0\}\}?$
Let $p$ be prime. Consider the group $G$ given by the presentation
$$P=\langle a,b,\{ x_m\mid m\in\Bbb N\}\mid a^p, \{x_m^{p^m}, x_m=b^mab^{-m}\mid m\in\Bbb N\}\rangle.$$
My question is two-fold:
Are ...
- 43.5k
0
votes
0
answers
38
views
Order of the group $\langle x, y \mid x^3 = y^2 = (xy)^3 = 1\rangle$. [duplicate]
I would like to determine the order of the group $G = \langle x, y \mid x^3 = y^2 = (xy)^3 = 1\rangle$.
Here is a proof that $G$ contains at most $12$ elements. Any element $g \in G$ can be ...
- 2,167
1
vote
1
answer
56
views
Find three subgroups of $QD16$ of order $8$, and find the isomorphism type of each group.
Consider the group of order $16$ with the following presentation:
$QD_{16} = ⟨σ, τ \mid σ^8 = τ^2 = 1, στ = τσ^3⟩$ Called the quasidihedral group of order $16$.
Find three subgroups of $QD16$ of ...
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2
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1
answer
70
views
The abelianization of $\langle t_1,...,t_n | t_1^2...t_n^2 \rangle$ is $\mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}$
Hello I want to prove that the abelianization of $G=\langle t_1,...,t_n | t_1^2...t_n^2 \rangle$ is $H=\mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}$.
In my textbook they display an inverse of the ...
- 367
0
votes
0
answers
42
views
a typo about free groups in Dummit's Abstract Algebra
I am not sure that if there is a typo in Dummit's Abstract Algebra on page219 : Let $S=G$ and the map $\pi:F(S)\to G$ is the homomorphism extending the identity map of $S$ . the first paragragh writes ...
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3
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1
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110
views
Prove that the group $G = \langle x, y \, \mid \, x^2=y^3=(xy)^3=1 \rangle$ is finite.
Let $G = \langle x, y \, \mid \, x^2=y^3=(xy)^3=1 \rangle$. I am trying to prove that $G$ is finite and list all the elements.
Using the fact that $(xy)^2=(xy)^{-1}=y^2x$, I have managed to show that ...
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1
vote
0
answers
72
views
Maximal quotient group of direct product
In GAP small group library, The group $$[32,2]=\langle a,b,c\mid a^4=b^4=c^2=1, ba=abc, [a,c]=[b,c]=1 \rangle.$$
We say $G/N$ is a maximal quotient group if there exists no quotient group $G/K$ such ...
- 53
2
votes
1
answer
59
views
relationship between Torsion and Relations [closed]
We know that every abelian group $H$ has a presentation of the form $\langle S |R\rangle$, where $S$ are the generators and $R$ are the relations.
Intuitively there should be some connection between $...
0
votes
1
answer
110
views
Definition of Presentation of Group in Dummit’s Abstract Algebra [closed]
A subset $S$ of elements of a group $G$ with the property that every element of $G$ can be written as a (finite) product of elements of $S$ and their inverses is called a set of generators of $G$. We ...
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2
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91
views
What is the order of a group given three generators, $a^2=b^3=c^4=1$ and $cb=ac$
A group $G=\langle a,b,c\mid a^2=b^3=c^4=1, cbc^{-1} = a\rangle $ what is the order of the group $G$ give all such possible values.
My attempt:
Since $cbc^{-1} =a \Rightarrow cb = ac\;\;(*)$, but then ...
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3
votes
1
answer
152
views
A basic question on group presentation
This is likely a pretty basic question, but I couldn't find answers anywhere. It's to do with how group presentations are represented.
It's said that, for example, the cyclic group of order $n$ may be ...
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4
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1
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77
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Minimal presentation of non-abelian group of order $p^3$ and exponent $p^2$
Let $p$ be an odd prime and let $H$ be the non-abelian $p$-group of order $p^3$ and exponent $p^2$. A presentation of $H$ is given by
$$H=\langle a,b \mid a^{p^2}=1,b^p=1,[a,b]=a^p \rangle. $$
How do ...
- 1,026
0
votes
1
answer
68
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Unique factorization of elements in $SL_2(\mathbb{Z})$
It is well know that the group $SL_2(\mathbb{Z})$ is generated by the matrices $S= \begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$ and $T= \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$. ...
- 105
1
vote
2
answers
71
views
Is the following presentation for $D_8$ a valid presentation?
My professor used the following presentation for $D_8:$
$$D_8 = \{s,t | s^2 = t^2 = (st)^4 = 1\}$$
But I am not sure if this presentation is correct, I looked at subWiki here https://groupprops....
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7
votes
2
answers
306
views
Why is $\langle a,b,c,d\mid abcda^{-1}b^{-1}c^{-1}d^{-1}\rangle\cong \langle a,b,c,d\mid aba^{-1}b^{-1}cdc^{-1}d^{-1}\rangle$?
We have two ways to present a surface of genus 2: (see Identifying the two-hole torus with an octagon)
Now, if one were to calculate the fundamental groups of the above surfaces using van Kampen's ...
- 1,524
0
votes
1
answer
79
views
Show a representation is faithful
I am working with the dihedral group $D_{8}$ of order $8$, given by the presentation
$$D_{8} = \left\langle a,b: a^{4}=b^{2}=1,\: b^{-1} ab = a^{-1}\right\rangle.$$
I am trying to find if this ...
- 2,704
2
votes
0
answers
48
views
Why are groups of type $F_2$ finitely presented?
There are some equivalent definitions for "finiteness properties", but let's define that $G$ is a group of type $F_n$ if it is the fundamental group of a CW complex $X$ whose $n$-skeleton is ...
- 91
1
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1
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112
views
How can one figure out if equations in presentation of a group are independent or not?
Note: My question is not directly about solving the below problem.
Let $G$ be a group with presentation given by $$G= \langle a,b,c \mid ab =c^2a^4,bc=ca^6,ac=ca^8,c^{2018}=b^{2019}\rangle.$$ ...
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8
votes
1
answer
125
views
In the group $G=\langle r,s,t\mid r^2=s^3=t^3=rst\rangle,$ the element $rst$ has order $2$
Formally, if $F$ is the free group with basis $X = \{r, s, t\}$ and $N$ is the normal subgroup generated by $R = \{r^2 s^{-3}, s^3 t^{-3}, t^{3} (rst)^{-1}\}$, and $G = F/N$, I want to show that the ...
- 465
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0
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81
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Are there nonabelian groups $G$ s.t. $G=\langle a_1,\dots,a_{n-1}\mid a_i^2=\epsilon,\epsilon^2=1,a_ia_j=\epsilon a_ja_i \textrm{ for }i\ne j\rangle.$
I study a proof of Hurwitz's theorem on bilinear compositions of sums of squares. The proof relies on the following lemma (which is given as an exercise):
Lemma. Let $G=\langle a_1,\dots,a_{n-1}\mid ...
- 1,303
1
vote
2
answers
114
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$\langle a,b\mid ab=1\rangle\cong \mathbb{Z}$
I'm new to free groups and presentation of groups, and I am having some problems with some basic facts:
Let $\langle a,b\mid ab=1\rangle$ be a group presentation. I want to show that $\langle a,b\mid ...
- 753
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votes
1
answer
98
views
Step in a proof that $G:=\langle x,y \mid x^n,y^2, (xy)^2\rangle$ is isomorphic to the dihedral group $D_n$
I’m reading a proof of the fact that the group given by the presentation $G:=\langle x,y \mid x^n,y^2, (xy)^2\rangle$ is isomorphic to the dihedral group $D_n$. It begins like this:
Let $G=\langle \...
- 411
1
vote
1
answer
102
views
Prove that $\langle a,b\mid ab=ba^n\rangle$ is not isomorphic to $\langle a,b\mid ab=ba^{n'}\rangle$ when $n\neq n'$
For a positive integer $n$, let $X_n$ be be the quotient space $\left([0,1] \times S^1\right) / \sim$, where the equivalence relation $\sim$ is generated by
$$
(0, z) \sim\left(1, z^n\right), \quad \...
- 865
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votes
1
answer
61
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Amalgamated product of finite cyclic groups is not abelian
I want to show that in general the amalgamated product of finite cyclic groups is not abelian.
Therefore, I define
$A := \langle a \rangle$, $H_1 := \langle x \mid x^2 \rangle$ and $H_2 := \langle y \...
- 169
1
vote
2
answers
110
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Suppose $G=\left\langle x, y, t\mid x^7=y^7=t^3=1, txt^{-1}=x^2, tyt^{-1}=y\right\rangle$. Show that $y\in Z(G)$.
The Problem: Suppose $G=\left\langle x, y, t\mid x^7=y^7=t^3=1, txt^{-1}=x^2, tyt^{-1}=y\right\rangle$. Show that $y\in Z(G)$.
My Attempt: Clearly $y$ commutes with $t$, so $y$ commutes with $t^2$ as ...
- 1,179
0
votes
2
answers
92
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Are group presentations useful in studying the pure math of a group as a whole?
Group presentations are often used by computers to quickly identify an element of a group or multiply two elements. However, they don't give a full understanding of the mathematical properties of a ...
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2
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0
answers
183
views
A group presentation given by the Collatz sequence of a fixed $n$. Has this been studied before?
This is a reference-request question.
The Set Up:
Fix $n\in\Bbb N$.
Suppose
$$C(x)=\begin{cases}
x/2&:x\text{ is even},\\
3x+1&:x\text{ is odd}.
\end{cases}$$
Let $m\in\Bbb N\cup\{0\}$. Denote ...
- 43.5k