Tagged Questions
0
votes
1answer
41 views
free groups question
I need help to solve this question
If $f:G_1\to G_2$ and $g:H_1\to H_2$ are homomorphisms of groups, then
there is a unique homomorphism $h:G_1*H_1\to G_2*H_2$ such that
$h_{G_1}=f$ and ...
2
votes
4answers
83 views
Every nonidentity element in a free group $F$ has infinite order
I'm trying to prove that every nonidentity element in a free group $F$ has infinite order. I'm really new on free groups and I found this subject really strange I couldn't understand it very well yet, ...
3
votes
0answers
64 views
Combinatorial Group Theory / Lyndon & Schupp, Proposition 4.1.. Automorphisms of free groups
I have problem to understand the three final lines of the proof of the proposition 4.1 in page 23.
The proposition is a generalization of the fact that $Aut(F_{n})$ is f.g.
It says that if $F$ is a ...
6
votes
2answers
42 views
Free groups and derivative
Per definition a derivative on a group $G$ is a mapping $D:G\rightarrow\mathbb{Z}G$ such that $D(gh)=D(g)+gD(h)$. Now my question:
uppose $G$ is a free group $F=F(X)$ with $X$ a finite set and suppose ...
4
votes
1answer
46 views
Lower central series of a free group
Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding ...
1
vote
2answers
75 views
$\frac{\mathbb{Z}*\mathbb{Z}}{N} \cong \mathbb{Z}\oplus\mathbb{Z}$
If $N$ is the normal subgroup of $\mathbb{Z}*\mathbb{Z}$ generated by $aba^{-1}b^{-1}$. How to show that $\frac{\mathbb{Z}*\mathbb{Z}}{N} \cong \mathbb{Z}\oplus\mathbb{Z}$?
3
votes
2answers
90 views
Must a surjection $F_2\to F_2$ also be an injection?
In other words, are there words $w_1$ and $w_2$ in the free group $F_2$ such that $F_2=\langle w_1, w_2\rangle$ but which are not a free basis for $F_2$?
I'm sure I'm missing a simple argument or a ...
1
vote
1answer
56 views
Proof that every subgroup of a free group is free (without using algebraic topology)
The fact that every subgroup of a free group is free follows quite naturally from the theory of covering spaces applied to graphs.
Is it also possible to proof this theorem without topology, by using ...
2
votes
1answer
37 views
lower central series and basic generators
After reading the following question:
Third quotient of the lower central series
I have a similar question.
Let us denote $F_n$ the free group with $n$ generators, $x_1,...,x_n$ and let $[a,b] = ...
9
votes
4answers
185 views
Is the free group on an empty set defined?
I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.
1
vote
3answers
93 views
Question on alphabet of a free group and its generating set
A free group (denoted by $F(S)$) on a set $S$, called the alphabet of the free group is set of all concatenations, of the elements of $S$ and its inverses (called alphabets). In some places, I have ...
2
votes
3answers
90 views
The free group $F_2$ contains $F_k$
I want to prove the following: the free group $F_2$ contains the free group $F_k$ for every $k \geq 3$. I am wondering whether the following line of reasoning is correct or not:
Suppose that $\lbrace ...
1
vote
1answer
58 views
Question related to commutator collection process and free groups
Let $F$ be a free group, generated by $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$. Assume that $n > 1$. For every $i$, let $\varphi_i: F \rightarrow F$ be the endomorphism of $F$ defined by ...
10
votes
1answer
167 views
When is $G \ast H$ solvable?
In a proof that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ is not finitely presented, I showed that $\mathbb{Z}_2 \ast \mathbb{Z}$ is not solvable. More precisely, one can prove that the ...
7
votes
3answers
133 views
Free group n contains subgroup of index 2
My problem is to show that any free group $F_{n}$ has a normal subgroup of index 2. I know that any subgroup of index 2 is normal. But how do I find a subgroup of index 2?
The subgroup needs to have ...
3
votes
1answer
117 views
Calculating the index of a subgroup in a free group
Can someone help me with the following question?
Let $F=\langle x_1,\ldots,x_n \rangle $ be a finitely-generated free group .
Let $p$ be a fixed prime, and $H$ be the normal subgroup of $F$ ...
0
votes
1answer
128 views
How can I show the free abelian group of rank $r$ is isomorphic to an $r$-copy of $\mathbb{Z}_\infty$?
Today, this problem was given to me.
Let $F$ be a abelian free group of rank $r$. Show that it is isomorphic to an $r$-copy of $Z_{\infty}$.
I could do some messy job about it but so far I ...
5
votes
1answer
58 views
Permutation Group as quotient of Free Group
I could use some help with the following question:
Let $S_{n}$
be the permutation group of $\left\{ 1,...,n\right\}$
, what is the minimal $k\in\mathbb{N}$
such that $S_{n}$
is a quotient of ...
3
votes
2answers
88 views
free group generated by polynomials
Someone recently asked me how to proof that $x+1$ and $x^3$ generate a free group. A colleague has worked out a proof. I have a vague memory that this has been studied, maybe a Monthly problem? Does ...
27
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
3
votes
1answer
163 views
Why isn't this free product of groups abelian?
I'm trying to prove that the free group $A=A_1*A_2$, where $A_1, A_2\neq 1$ is not abelian. Following the hints below:
Let $x,y\in A_1*A_2$, where $x\neq y$.
Suppose now $A_1=F(S)$ and $A_2=F(T)$, ...
7
votes
2answers
146 views
What good are free groups?
In Algebra: Chapter 0, one learns two definitions of Free Groups associating with sets.
Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where ...
0
votes
1answer
86 views
Free Groups and their Generators - Specific Question
1) Let $F$ be the free group on the three generators $x,y,z$. For non-zero integers $r,s,t$
then CLAIM: the subgroup of $F$ generated by $x^r , y^s , z^t$ is freely generated by these elements.
2) ...
2
votes
0answers
89 views
In a free group two elements commute if and only if they are powers of a common element
In other words, $uv = vu$ in $F_n$ if and only if $u=w^m$ and $v=w^n$ for some $w\in F_n$.
I would like to prove this without making use of Nielsen-Schreier (every subgroup of a free group is free). ...
0
votes
3answers
101 views
Let $x$ , $y$ be the basis of a free abelian group of rank 2, prove that $2x+3y$ and $ x-y$ generate a free subgroup
The problem here is that the matrix
$$\begin{pmatrix}
2 & 3 \\
1 & -1 \\
\end{pmatrix}$$
is invertible but not unimodular and hence the elements $2x+3y$ and $x-y$ generate a free abelian ...
3
votes
1answer
86 views
Maximal Free Subgroups and Torsion
Let $G$ be a non-trivial torsion-free Abelian group.
If $T$ is a maximal free subgroup of $G$, then $G\over T$ is periodic?
How can we prove this?
0
votes
2answers
68 views
Proof of Existence of a Free Group (one particular step in the process)
I'm working through a lemma which is used to prove the existence of a free group on a set $S$.
The setting is this:
$S$ is a set, and $G$ is a group, and $f:S\to G$ is a map such that the image of ...
2
votes
1answer
330 views
Commutator subgroup of a free group
Let $F_k$ be the free group of rank $k$.
If $k=2$ it is not hard to see that the set $\{[s_1^{n_1},s_2^{n_2}] \mid n_i\neq 0\}$ is a basis for $F_2'$. (Prime denotes the commutator subgroup).
What ...
2
votes
2answers
93 views
Index Of Commutator Subgroup In Free Groups
Given a free group of rank $n$ -> $F_n$ .
Is it easy to see what is the index of the subgroup $ [F_n, F_n ] \subseteq F_n $ ?
Hope someone will be able to help me understand this
Thanks !
3
votes
2answers
122 views
A free group on the non-empty set $X$ is solvable iff $|X| =1$
Let $X$ be a non-empty set. Prove that $F_X$, the free group on $X$ is solvable if and only if $|X| = 1$.
We can see that if $|X| = 1$, then $F_X$ is abelian, and hence solvable. However, the other ...
0
votes
2answers
119 views
Let free group, F and F/F' have same rank
Let F be a free group of rank n>1.
Then F and F/F' have same rank.
Please help me!
2
votes
1answer
102 views
Centralizers in free metabelian groups
My question was inspired by the recent post:
The centralizer of an element x in free group is cyclic.
Is it true that non-identity elements have abelian centralizer in free metabelian groups?
...
3
votes
3answers
242 views
Quotient of two free abelian groups of the same rank is finite?
Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B
]$ (the index of $B$ in $A$) is finite.
For ...
3
votes
1answer
92 views
Presentation of $A_n$, from Jacobson's Basic Algebra I.
I want to show $A_n$ can be defined by the following relations $x_1,x_2,\dots,x_{n-2}$:
$$
x_1^3;\qquad x_i^2, i>1;\qquad (x_ix_{i+1})^3;\qquad (x_ix_j)^2, j>i+1.
$$
I assume $n\geq 3$, and let ...
7
votes
2answers
220 views
Group isomorphism concerning free group generated by $3$ elements.
From Jacobson's Basic Algebra I, page 70,
Let $G$ be the group defined by the following relations in $FG^{(3)}$:
$$x_2x_1=x_3x_1x_2, \qquad x_3x_1=x_1x_3,\qquad x_3x_2=x_2x_3.$$
Show that $G$ ...
3
votes
0answers
147 views
Double Coverings of the Double Torus
I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is
$$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$
where ...
0
votes
2answers
82 views
Free group equivalent definition
I am trying to prove that group P is free if and only if it satisfies the following:
for any groups $A,B$, any homomorphism $\pi: P \rightarrow B$ and any epimorphism $\alpha: A \rightarrow B$ there ...
2
votes
1answer
267 views
Free group and universal property
I'm trying to understand universal properties. An example is the definition of a free group (as I understand it so far):
Revised definition:
A free group $F_S$ over a set $S$ is a pair $(g,F_S)$ ...
0
votes
1answer
56 views
Regular normal forms of two groups
If ${\bf G}$ and ${\bf H}$ have regular normal forms, then so does ${\bf G}$ x ${\bf H}$ and ${\bf G}$ * ${\bf H}$.
I am new to the concepts of regular normal forms. If anyone could offer any ...
3
votes
2answers
124 views
Question on the presentation of $(\mathbb{R}, +)$
In this question, it is shown that $(\mathbb{R}, +)$ is not a free group. But my question is: if it is not a free group, exactly what relations is it subject to?
My other question is: are there ...
3
votes
0answers
106 views
A question about bases of free groups
Let $F_n$ be a free group of finite rank $n$ and let $\gamma_m(F)$ denote the $m$-th term of the lower central series of $F$ where $m \ge 1$ is a natural number. Suppose that $x,y$ are primitive ...
0
votes
0answers
56 views
The free group on 3 generators is isomorphic to a subgroup of the free group on 2 generators. [duplicate]
Possible Duplicate:
Is there a free subgroup of rank 3 in $SO_3$?
How is this possible?
Here are some facts:
$F_2 \subset F_3$
$F_2 \not\cong F_3$
Fact 2 implies that $F_3$ is ...
3
votes
2answers
115 views
Normal subgroup of automorphisms of a free group
Let $F_2=\langle X,Y\rangle$ be the free group of rank $2$ and consider $A,B,C\in Aut(F_2)$ given by: $$A(X,Y)\mapsto(YX^{-1}Y^{-1},Y^{-1})$$ $$B(X,Y)\mapsto(X^{-1},Y^{-1})$$ ...
3
votes
4answers
178 views
$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$
I am confused by the proof a proposition:
$F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$
The proof is:
...
6
votes
2answers
312 views
Free group as a free product
Let $G$ be a group generated by two elements $a$ and $b$.
Suppose $G$ is a free group of rank 2.
Is it true that $G=\langle a\rangle * \langle b\rangle$?
I think the problem is that the definitions I ...
4
votes
0answers
160 views
Identifying a certain subgroup of a free group
Let $F$ be the free group on the generators $a_1,\ldots,a_n$. Define homomorphisms $\phi_i:F\to F$ by $\phi_i(a_j)=a_j^{1-\delta_{ij}}$, where $\delta_{ij}$ is the Kronecker delta; basically, $\phi_i$ ...
2
votes
1answer
147 views
Quotient of free group with normal subgroup
I'm trying to make up an example of a quotient of a free group to check if I understand quotients properly. I do for the usual cases but I've not seen free groups before. So here I go:
Let $F = ...
4
votes
1answer
80 views
A lemma about free groups
Let F be a finitely generated free group and $\gamma_m$ the lower central series.
Why is $\gamma_m(F)/\gamma_{m+1}(F)$ torsionfree? I know it is abelian, but I couldn't find out more about it, as ...
1
vote
1answer
262 views
Free groups and commutators
Good evening
I was trying to prove that the commutator [F2,F2] of the free group F2 is not finitely generated by using covering spaces (i have to admit that this is the idea of a friend) it seems ...
4
votes
4answers
544 views
Generators of a free group
If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?
