All Questions
Tagged with free-groups finitely-generated
27
questions
2
votes
0
answers
37
views
How to prove the free group $F_n$ cannot be generated by $n$ elements as a monoid?
I cannot figure out how to prove that the submonoid of $F_n$ generated by elements $\alpha_1,\dots , \alpha_n$ will never be the whole group.
It clearly is possible to generate the free group on $a_1, ...
- 5,623
2
votes
0
answers
51
views
Equations on Free Groups
I was trying to read an old paper concerning equations on free groups and immediately came to a puzzling statement which made me wonder whether I am fundamentally misunderstanding something or if I am ...
- 990
2
votes
1
answer
54
views
What is the most concise way to present a particular subgroup of $F_2$?
Let $A\leq F_2$ where $F_2$ is the free group on $\{a,b\}$.
Assume $A$ is generated by words on $a,b$ such that the total powers of $a$ are $3x$ for some $x\in \mathbb{Z}$ and the total powers of $b$ ...
- 2,454
0
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0
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47
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Is there any proof of $\#(F/N)=2n$ which doesn't use any group other than $F/N$ itself? (Michael Artin "Algebra 1st Edition")
I am reading "Algebra 1st Edition" by Michael Artin.
The following proposition is Proposition (8.3) on p.221 in this book.
(8.3) Proposition. The elements $x^n,y^2,xyxy$ form a set of ...
- 8,740
0
votes
0
answers
22
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Question about a proposition about free groups, generators and relations. Is it true or false that $N=\ker\phi$ holds? Michael Artin "Algebra 1st Ed."
I am reading "Algebra 1st Edition" by Michael Artin.
I feel free groups, generators and relations are very difficult.
The following proposition is Proposition (8.3) on p.221 in this book.
(...
- 8,740
4
votes
2
answers
172
views
What is the kernel of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$?
What is the kernel $K\leq F_2 = \langle a,b \rangle$ of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$ given by $a \mapsto (1+2\mathbb{Z},0+3\mathbb{Z})$ and $b\mapsto (0+2\mathbb{Z}, 1+3\...
- 2,454
1
vote
1
answer
200
views
A group is locally free exactly when its finitely generated subgroups are free
This is Exercise 6.1.9 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search for "locally free finitely generated" in the group theory tag, ...
- 45.7k
2
votes
1
answer
121
views
Endomorphism monoid of free group
The automorphism group of $F_2$(the free group generated by two elements) is finitely generated. Especially, it is generated by Nielsen transformations. If we consider all homomorphisms from $F_2$ to ...
- 150
4
votes
1
answer
305
views
Whether a group containing a free group is also a free group
Suppose $G$ is a group generated by two elements $s,t$.
Suppose $H$ is a subgroup of $G$ such that $H= \langle s^k,t^k \rangle$ is a free group where $k$ is some integer not equal to $1$.
Does it ...
- 361
1
vote
1
answer
55
views
Free group on a set B. If the free group is finitely generated, then B is finite.
Let B be a set. For simplicity, assume B contains all the formal inverses of its elements. Let W(B) be the set of words created from elements in B, and let F(B) be the set of equivalence classes [w] ...
- 377
1
vote
1
answer
71
views
Are all faithful actions of finite rank free groups ping-pong actions?
Suppose, $G$ is a finitely generated group with a finite set of generators $A$. Suppose $G$ is acting on a set $S$. Let’s call such action a ping-pong action iff $\exists$ a collection of pairwise ...
- 15.7k
4
votes
1
answer
49
views
Subgroup generated by commensuration class of an element of a virtually free group
Suppose $G$ is a group. Let’s call $a, b \in G$ commensurate, iff $\exists m, n \in \mathbb{Z} \setminus \{0\}$, such that $a^n = b^m$. One can see, that commensuration is an equivalence relationship:
...
- 15.7k
1
vote
0
answers
16
views
Is there some sort of formula for $[F_n : V_{\{x^4\}}(F_n)]$?
Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_4(n) = [F_n : V_{\{x^4\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_4(n)$? ...
- 15.7k
1
vote
1
answer
420
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Show that a group $G$ is finitely generated if and only if it is a quotient of a free group on a finite set of letters
As stated in the title, I am working on this exercise:
Show that a group $G$ is finitely generated if and only if it is a quotient of a free group on a finite set of letters.
I have showed $(\...
- 5,930
0
votes
1
answer
37
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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.
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