Questions tagged [free-groups]
Should be used with the (group-theory) tag. Free groups are the free objects in the category of groups and can be classified up to isomorphism by their rank. Thus, we can talk about *the* free group of rank $n$, denoted $F_n$.
962
questions
4
votes
2
answers
605
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 ...
- 8,823
3
votes
0
answers
179
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 ...
- 63
10
votes
1
answer
6k
views
Is there a simple proof of the fact that if free groups $F(S)$ and $F(S')$ are isomorphic, then $\operatorname{card}(S)=\operatorname{card}(S')?$ [duplicate]
Theorem. If $F(S)$ and $F(S')$ are isomorphic free groups with bases $S$ and $S'$ respectively, then $\operatorname{card}(S)=\operatorname{card}(S').$
I know a proof of this fact that ...
user23211
1
vote
0
answers
107
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 isomorphic to ...
- 836
3
votes
3
answers
1k
views
Proving that a group generated by x,y and z and a given relation is actually free
I'm trying to show that a group generated by elements $x,y,z$ with a given relation $xyxz^{-2}=1$ (where $1$ is the identity) is in fact a free group.
What are some usual ways of going about this ...
- 1,439
4
votes
2
answers
416
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: $$A(X,Y)\mapsto(YX^{-1}Y^{-1},Y^{-1})$$ $$B(X,Y)\mapsto(X^{-1},Y^{-1})$$ $$C(X,Y)\mapsto(X^{-1},XY^...
- 2,179
4
votes
4
answers
651
views
$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$
I am confused by the proof a proposition:
$F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$
The proof is:
Let $...
- 7,926
11
votes
2
answers
3k
views
Free group as a free product
Let $G$ be a group generated by two elements $a$ and $b$.
Suppose $G$ is a free group of rank 2.
Is it true that $G=\langle a\rangle * \langle b\rangle$?
I think the problem is that the definitions I ...
- 113
6
votes
0
answers
236
views
Identifying a certain subgroup of a free group
Let $F$ be the free group on the generators $a_1,\ldots,a_n$. Define homomorphisms $\phi_i:F\to F$ by $\phi_i(a_j)=a_j^{1-\delta_{ij}}$, where $\delta_{ij}$ is the Kronecker delta; basically, $\phi_i$ ...
- 7,154
3
votes
1
answer
720
views
Quotient of free group with normal subgroup
I'm trying to make up an example of a quotient of a free group to check if I understand quotients properly. I do for the usual cases but I've not seen free groups before. So here I go:
Let $F = \...
- 46.8k
4
votes
1
answer
142
views
A lemma about free groups
Let F be a finitely generated free group and $\gamma_m$ the lower central series.
Why is $\gamma_m(F)/\gamma_{m+1}(F)$ torsionfree? I know it is abelian, but I couldn't find out more about it, as ...
- 5,432
1
vote
1
answer
897
views
Free groups and commutators
Good evening
I was trying to prove that the commutator [F2,F2] of the free group F2 is not finitely generated by using covering spaces (i have to admit that this is the idea of a friend) it seems ...
- 1,571
14
votes
4
answers
4k
views
free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$
I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication (...
- 10.7k
8
votes
3
answers
1k
views
Alternative "functorial" proof of Nielsen-Schreier?
There are two proofs of Nielsen-Schreier that I know of. The theorem states that every subgroup of a free group is free. The first proof uses topology and covering space theory and is rather elegant. ...
user2055
6
votes
4
answers
3k
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?
- 929