All Questions
Tagged with free-groups group-presentation
61
questions
3
votes
1
answer
54
views
Prove a surjective endomorphism $\phi$ of a 1-relator group $ ⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩ $ is not injective
Consider the infinite group $H$ with presentation
$$
⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩
$$
so that the relation is $a^{-1} b^2 a=b^3$.
The map
$$
a ↦ a\\b ↦ b^2
$$
induces a surjective homomorphism $ϕ:H\to ...
- 3,047
-1
votes
1
answer
65
views
Can you determine the order of a generator in this group presentation? [closed]
Given the following group presentation $<x,y|2x+3y=0, 5x+2y=0>$ of an Abelian group, find the order of element x.
My follow up question: Is there a way to determine the order without finding ...
- 140
2
votes
0
answers
107
views
Can the following proof calculus show that any finitely presented free group is free?
If a finitely presented group is free, will it always have a proof in the proof calculus outlined in this question that it is free?
I recently saw this question.
I tried to show that the group was ...
- 11.9k
2
votes
0
answers
55
views
Presentation of Product Group
Here is the question I have been working on:
If $G_1 = \langle X_1 : R_1\rangle$ and $G_2 = \langle X_2 : R_2\rangle$, supply a presentation for $G_1 \times G_2$.
Deduce that, if $G_1$ and $G_2$ are ...
13
votes
2
answers
204
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 ...
- 623
1
vote
0
answers
150
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 $\pi_1(K)$, ...
- 592
0
votes
0
answers
49
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 ...
- 351
8
votes
1
answer
142
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 ...
- 619
1
vote
2
answers
130
views
$\langle a,b\mid ab=1\rangle\cong \mathbb{Z}$
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 ...
- 763
0
votes
2
answers
135
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 \...
- 1,788
2
votes
0
answers
132
views
Universal Property of the Free Abelian Group on $S$.
This problem is from Dummit and Foote, 6.3.11.
Problem: Let $S$ be a set. The group with presentation $(S,R)$ with $R = \{[s,t] \mid s,t \in S\}$ is called the free-abelian group on $S$ denoted $A(S)$....
- 2,673
2
votes
0
answers
52
views
Finding relators of a matrix group
Let $f_1,\dots,f_n$ be maps from $\mathbb{R}$ to $\mathbb{R}$ of the form $f_i(x) := a_ix + b_i$ with $a_i,b_i \in \mathbb{Q}$. We construct the transformation group $G = \langle f_1, \dots, f_n \...
- 203
1
vote
0
answers
129
views
Presentation of the amalgamated product of $G_1$ and $G_2$ above $H$ is $ \langle S_1,S_2\; ; \; R_1,R_2,\phi_1(s)\phi_{2}^{-1}(s),s \in S \rangle $
Here, I found the proof of presentation of the free product of groups. I wanted to show the same thing for amalgamated free product of 2 groups i.e.
Show that that if $H$ is generated by $S$, and $ \...
- 1,377
3
votes
1
answer
128
views
For any primitive element $a$ in a free group of rank two we have $a^k ba^l=b$ only if $(k,l)=(0,0)$ provided $b\not\in \langle a\rangle$
Problem 1: Let $F$ be a free group of rank at least two, and $a, b\in F$ be two
non-trivial elements with $b\not \in \langle a\rangle$. Suppose, $a\neq x^n$ for any $x\in F$ and any
$n\geq 2$, i.e. $...
- 493
2
votes
1
answer
220
views
S_3 is a quotient of the free group F({x,y})
I am self-learning Algebra: Chapter 0 by Paolo Aluffi. He defined a presentation of a group $G$ as follows:
So, according to my understanding, $R$ is the kernel (a normal subgroup) of the surjection $...
- 602