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 $F_n$.
101
questions
24
votes
5
answers
8k
views
Abelianization of free group is the free abelian group
How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
user88576
21
votes
4
answers
10k
views
Commutator subgroup of rank-2 free group is not finitely generated.
I'm having trouble with this exercise:
Let $G$ be the free group generated by $a$ and $b$. Prove that the commutator subgroup $G'$ is not finitely generated.
I found a suggestion that says to ...
- 6,955
21
votes
6
answers
3k
views
Different ways of constructing the free group over a set.
This could be too broad if we're not careful. I'm sorry if it ends up that way.
Let's put together a list of different constructions of the free group $F_X$ over a given set $X$.
It seems to be ...
- 45.7k
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
1
answer
2k
views
Prove that $PSL(2,\mathbb{Z})$ is free product of $C_2$ and $C_3$
Prove that $PSL(2,\mathbb{Z})=C_2 \star C_3$.
Now $C_2 \star C_3=\langle a,b\ |\ a^2, b^3 \rangle$ i.e. the free product.
But how do I show that presentation of $PSL(2,\mathbb{Z})$ is this?
- 11.4k
14
votes
1
answer
749
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 ...
- 33.3k
9
votes
4
answers
4k
views
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$ [duplicate]
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$.
My try:
Let $\mathbb Z^n\cong \mathbb Z^m $. To show that $m=n$.
Case 1: Let $m>n$. Now that $\mathbb Z^m$ has $m$ ...
- 31.2k
9
votes
4
answers
3k
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 ...
- 113
25
votes
1
answer
3k
views
What is tricky about proving the Nielsen–Schreier theorem?
The Nielsen–Schreier theorem states (in part):
Let $F$ be a free group, and $H\le F$ be any subgroup. Then $H$ is isomorphic to a free group.
I have seen the topological proof of this theorem ...
- 6,828
5
votes
3
answers
821
views
Homomorphism between free groups
Let $F_a$ be the group which is freely generated by $a$ elements. How to show that there is a homomorphism from $F_a$ onto $F_b$ if and only if $b\le a$?
I was thinking one possibility is if $F_a$ ...
- 71
20
votes
3
answers
10k
views
Finding subgroups of a free group with a specific index
How many subgroups with index two are there of a free group on two generators? What are their generators?
All I know is that the subgroups should have $(2 \times 2) + 1 - 2 = 3$ generators.
letiitbe
14
votes
3
answers
1k
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 ...
- 15.1k
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
9
votes
1
answer
735
views
Slicker construction of the free product of groups
The usual construction of the free product of groups $\{G_i\}_{i \in I}$ consists of taking the discriminated union $\coprod_{i \in I} G_i$ and taking the set of words satisfying a handful of ...
- 1,057
8
votes
1
answer
453
views
Does this group presentation define a nontrivial group?
Given a presentation
$$
\langle x,y,z : x^y=x^2, y^z=y^2, z^x = z^2 \rangle,
$$
where $x^y$ is just the usual conjugation (that is, $x^y$ is defined to be $y^{-1} xy$). Can we say for sure, whether ...
- 765