All Questions
Tagged with free-groups quotient-group
17
questions
4
votes
1
answer
102
views
Order of $xy$ in various quotients of the free product $G_1 * G_2$, where $x \in G_1, y \in G_2$ are nontrivial
Say we have two arbitrary nontrivial groups $G_1$ and $G_2$, and some arbitrary nontrivial elements $x \in G_1, y \in G_2$. Then it is known that the order of $xy$ in the free product $G_1 * G_2$ is ...
- 619
3
votes
1
answer
65
views
If $F$ is free and $R$ is normal in $F$, then $F/R'$ is torsion-free, where $R'=[R,R]$
If $F$ is free and $R$ is normal in $F$, then $F/R'$ is torsion-free, where $R'=[R,R]$.
This is Exercise 11.50 in Rotman's An Introduction to the Theory of Groups with the following hint attributed ...
- 619
0
votes
1
answer
131
views
Showing $F_X \cong F_Y\implies |X| = |Y|$ [duplicate]
Lately I've been studying free groups, I'm a layman on the subject but I came across a step in the demonstration that I couldn't move forward. I know the question seems to be good but: If $F_X \cong ...
- 3
0
votes
0
answers
68
views
Isomorphism of quotient groups: if $G_1 \cong G_2$, then $G_1/K_1 \cong G_2/K_2$?
Well, I have a little doubt. If $G_1$ and $G_2$ are free groups in $H_1$ and $H_2$ respectively. If we have $\langle x^2 : x \in G_1 \rangle = K_1 \lhd G_1$ and $\langle y^2 : y \in G_2 \rangle = K_2 \...
- 67
-2
votes
1
answer
47
views
If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$ [closed]
This is probably a basic fact of group theory but I am not able to prove it:
Let $F_2$ be the free group generated by 2 elements and $H,K$ be two normal subgroups of $F_2$. If $F_2/H$ is isomorphic ...
- 861
0
votes
0
answers
57
views
Order of a quotient of a free abelian group
Let $G\subseteq \mathbb{C}$ be a free abelian group of rank $n$ and let $p$ be a prime. Then, we know that $$|G/pG|=p^n$$ and in fact for this we don't even need $p$ to be a prime. Suppose now that ...
- 9
1
vote
0
answers
142
views
In $F=F^{ab}(A)$, define $f\sim f'$ if and only if $f-f'=2g$ for some $g\in F$. Show $F/\sim$ is finite if and only if $A$ is finite.
this is a question from Aluffi's Algebra: Chapter 0, Exercise II.5.10. The full question is the following:
Given a free abelian group over a set $A$, denoted $F=F^{ab}(A)$, we define an equivalence ...
- 51
6
votes
1
answer
127
views
Relations from quotient of free product
This question arose from an exercise which asks you show that the fiber coproduct exists in the category of groups. I was eventually able to (mostly) solve the problem by “gluing” the images of ...
- 563
0
votes
1
answer
108
views
Factor group of a free group
Let $F[A]$ be the free group on the generating set $A$. Let $C$ be the commutator subgroup of $F[A]$, then show that $F[A]/C$ is a free abelian group with basis $\{aC \mid a \in A\}$. It is trivial ...
- 711
1
vote
1
answer
72
views
Representing a group as a quotient of a free group
Consider $G=F \rtimes T$, where $F=\mathbb{Z}_3 \times \mathbb{Z}_3$ and $T=\mathbb{Z}_5$. Let $\phi : \mathbb{Z}_5 \rightarrow Aut(\mathbb{Z}_3 \times \mathbb{Z}_3)$.
It is said that any group is ...
- 43
0
votes
1
answer
267
views
Showing this quotient group is isomorphic to $\mathbb{Z}_2\times{}\mathbb{Z}_2$
How do I show that $\mathbb{Z}_4\times{}\mathbb{Z}_2/K$ is isomorphic to $\mathbb{Z}_2\times{}\mathbb{Z}_2$, where $K=\langle(2,0)\rangle$.
I'm getting confused with the details involved here, I will ...
- 1,285
0
votes
2
answers
489
views
How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?
Let $D_{2\cdot 8}$ be given by the group presentation $\langle x,y\mid xy = yx^{-1} , y^2 = e, x^8 = e\rangle$. Let $G = F_{\{x,y\}}$ be the free group on two generators and $N = \langle\{xyx^{-1}y,y^...
- 37.2k
4
votes
1
answer
111
views
Finding the quotient of this free abelian group
I have the group $\langle a,b,c\rangle/\langle -b+c-a,b+c-a\rangle$. I know this is $\mathbb{Z}\oplus\mathbb{Z_2}$. However, I tried doing it like this and got something else :
I have
$$-b+c-a=0, b+...
- 3,134
0
votes
1
answer
37
views
Suppose $F$ is a finitely generated free group, with $M,N \trianglelefteq F$ such that $M \leq N$. If $F/N \cong F/M$ then need we have $N=M$?
As the title states - suppose $F$ is some finitely generated free group, with $M,N \trianglelefteq F$ such that $M \leq N$. If $F/N \cong F/M$ then need we have $N=M$?
Thanks in advance.
- 742
1
vote
1
answer
67
views
Quotient of free groups $\langle a_1,\ldots, a_n\rangle /\langle a_2-a_1,a_3-a_2,\ldots,a_{n}-a_{n-1},a_1-a_n\rangle$
I am trying to show that the quotient
$$\frac{\langle a_1,\ldots,a_n\rangle}{\langle a_2-a_1,a_3-a_2,\ldots,a_{n}-a_{n-1},a_1-a_n\rangle}\cong \mathbb{Z}.$$
Is the following argument correct?
Since ...
- 11k