# 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 $F_n$.

677 questions
Filter by
Sorted by
Tagged with
26 views

### Representing a non-abelian group as a free group

Can we express the group $(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$, where $p,q$ are odd distinct primes, using a free group? Thanks a lot in advance.
65 views

### Quotient of a free group by a subgroup of the commutator

Let $F$ be the free group on finitely many generators $x_1,\dotsc,x_n$. Let $[F,F]\subseteq F$ be its commutator, so we know $F/[F,F]\cong \mathbb{Z}^n$, and let $N\subseteq F$ be a normal subgroup ...
36 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 ...
13 views

### Epimorphism from free product to direct product of groups.

Let $G_1$ and $G_2$ be two non abelian groups and $G_1 * G_2$ be the free product of these two groups. Can we define an epimorphism from $G_1 *G_2$ to the direct product $G_1 \times G_2$ of these ...
12 views

### Representation of free group

I want to prove that $$<a,b \ | \ aba^{-1}b^{-1},ab^{-1}ab>\cong \mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$$ I already show $a^2=1$, but I don't make sure that $a\neq1$. How can I prove this? Any ...
32 views

22 views

### Determining the cardinality of a system of free generators for the fundamental group of $E$.

This is exercise 3 from Munkres section 84. Let $X$ be the wedge of two circles; let $p:E\rightarrow X$ be a covering map. The fundamental group of $E$ maps isomorphically under $p_\ast$ onto a ...
27 views

### Properties unique to free groups

I was just working on a problem for my Algebra class about proving that a group was free. I was able to figure that out, but it got me wondering: are there any properties that are unique to free ...
34 views

67 views

### Showing $\mathbb{Z}\times\mathbb{Z}$ is isomorphic to the group $\langle x,y,|xy=yx\rangle$

Let $X=\{x,y\}$ such that $x\ne y$ and denote by $F(X)$ the free group constructed on $X$. Let $\phi_x$ and $\phi_y$ be the canonical injections of $\mathbb{Z}$ into $F(X)$. We have one relator: ...
38 views

40 views

### Free topological groups and quotients

I am learning about free topological groups, and I am trying to understand whether the analogue of "every group is a quotient of a free group" holds in the continuous setting too. I am particularly ...
46 views

39 views

### The counit of an adjunction of the adjoint pair free and underlying functor

I would like to know how the counit $\varepsilon$ for an adjunction $(F,U,\phi)$ $$\varepsilon:FUX\to X$$ works if $F$ is the free functor from $\mathbf{Set}$ to $\mathbf{Ab}$ and $U$ is the ...
39 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: ...
54 views

### Is there an algorithm that, given a finite presentation of a free group, outputs a free basis for the group?

I have been stuck on this problem for two weeks, and I'm not sure I believe it to be true anymore. I have no idea how to proceed. I know nothing about algorithms or how to find them, this is my first ...
42 views

### free group quotient by its commutator is a free abelian group via universal property

Let $F=F(A)$ be a free group, and let $f:A→G$ be a set-function from the set $A$ to an abelian group $G$. Show that $f$ induces a unique homomorphism $F/[F,F]→G$, where $[F,F]$ is the commutator ...
88 views

60 views

### A list of properties of the free group of rank two [closed]

Understanding the free group of rank two is (as far as I know) very important to many different problems. I thought it would be helpful (for me and for others) to know more about it. Can anyone give ...
48 views

### N contains the commutator subgroup if and only if G/N is abelian, proof by words?

Working on some problems after my first semester of abstract-algebra. I saw this question $\mathbf{Theorem}$ Let $N$ be a normal subgroup of a group $G$. Then $N$ contains the commutator subgroup ...
40 views

### Integral group ring of the free group on two generators

I know that the integral group ring $\mathbb Z[\mathbb Z]$ of $\mathbb Z$ is described as the ring of Laurent polynomials $\mathbb Z[t^{\pm}]$. I'm asking if there is a known description of the ...
69 views

### Here I prove every free group is free abelian. Where is the mistake?

Let $F$ be free over $X$. Then for any group $G$ and any $\alpha: X \to G$ there is a homomorphism $\beta: F \to G$ such that $\alpha = \beta|X$. Alright. Now, in particular, when $G$ is abelian $F$ ...
55 views

### Prove that $F_2$ is a subgroup of $F_3$, and construct a covering space of $S^1\vee S^1\vee S^1$ corresponding to this subgroup

As the title explains, I'm working on a question that asks me to prove that $F_2$ is a subgroup of $F_3$, and construct a covering space of $S^1\vee S^1\vee S^1$ corresponding to this subgroup. If ...
81 views

### Show that for each integer $n ≥ 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators. [duplicate]

I'm trying to answer a problem which asks me to show (using covering spaces) that for each integer $n ≥ 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators. ...
89 views

I am stuck at an exercise concerning a special subgroup of given free group. Let $F$ be free group, $n$ a fixed integer and $N = \langle R_n\rangle$ the subgroup generated by the set $R_n:=\{x^n: ... 0answers 61 views ### Show that$\langle a,b \mid aba^{-1}b^{-1} \rangle \cong \mathbb{Z}^2$. [duplicate] Show that$\langle a,b \mid aba^{-1}b^{-1}\rangle\cong \mathbb{Z}^2$. Proof Define$j: \{a,b\} \to \mathbb{Z}^2$via$j(a) = (1,0)$and$j(b) = (0,1)$. Define$i:\{a,b\} \to F(\{a,b\})$, where$F$... 1answer 86 views ### Show that$0 \to \operatorname{Ker} \partial \to C_n \to \operatorname{Im} \partial \to 0$splits. This comes from Hatcher's exercise 43 of section 2.2: (a) Show that a chain complex of free abelian groups$C_n$splits as a direct sum of subcomplexes$0 \to L_{n+1} \to K_n \to 0$with at most ... 1answer 52 views ### 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$(\...
This question arose from a statement from Munkres Section 69 that is seeming contradictory to his definition of free group. He defines the free group in the following way: Let $\{a_{\alpha}\}$ be ...