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 $F_n$.

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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.
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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 ...
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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 ...
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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 ...
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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 ...
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Show that the free group $F_n$ contains a subgroup isomorphic to $F_k$ whenever 1≤k≤n.

DEFINITIONS: A word in $X ∪ X^{−1}$ is an ordered set of $n ∈ N ∪ {0}$ elements, each from $X ∪ X^{−1}$, with repetitions allowed. We write a word in the following way: $w = x_{\lambda_1}^{\epsilon_1}...
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Products in the free group are associative

In Aluffi's Chapter 0 there is an exercise to show that the product in the concrete construction of a free group is associative by showing that the reduction of a word is independent of the order in ...
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31 views

symmetric power

Let F be a free R-module of rank r. Prove that $\mathbb{S}_R^l(F)$ is free. I know that this is talking about the $l^{th}$-symmetric power but how do I show it free.
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What does $\Phi\vert_S$ mean?

In Abstract Algebra (Dummit and Foote) Exercise 11 in Section 6.3 (A Word on Free Groups) it says: ... Prove that $A(S)$ has the following universal property: if $G$ is any abelian group and $\...
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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 ...
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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 ...
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Infinite Wedge Sum

Let $X$ be a topological space. $X$ is called Wedge of circles if $\exists \hspace{0.1cm} \left\lbrace S_{\alpha} \right\rbrace _{\alpha \in S}$ such that : (i) $S_{\alpha} \subset X \hspace{0.2cm} \...
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Proving that the elements of a free group satisfy a certain property

Let $X$ be a set and $F(X)$ the free group on $X$. Suppose $P(x)$ denotes some property. If I want to show that all the elements of $x\in F(X)$ satisfy $P(x)$, would it be enough to show that $P(x)$ ...
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101 views

Free group over a set of two elements is abelian

Let $A=\{a,b\}$ and $a\ne b$. Let $F(A)$ be the free group constructed on $A$. Let $f_a,f_b$ be the canonical homomorphisms of $\mathbb{Z}$ into $F(A)$. Let $g:F(A)\rightarrow\mathbb{Z}\times\mathbb{Z}...
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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: ...
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An isomorphism onto the additive group $\mathbb{Z}\times\mathbb{Z}$

Let $I=\{\alpha,\beta\}$ such that $\alpha\ne\beta$. Let $F(I)$ be the free group constructed on $I$ and $\phi_\alpha,\phi_\beta$ be the canonical injections of $\mathbb{Z}$ into $F(I)$. Write $r=\...
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When is a quotient group of a free group finite?

Let $I=\{\alpha\}$ and $k\in\mathbb{N}$. Consider the free group $F(I)$ constructed on $\{\alpha\}$. Let $\phi_\alpha$ be the canonical homomorphism of $\mathbb{Z}$ into $F(I)$. Let $r=\phi_\alpha(1)^...
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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 ...
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Free group generated by group

Let $G$ be a group, let $F(G)$ be the free group generated by $G$. Is it true that $F(G) \cong G$? By the universal property of free groups there exists a unique group homomorphism $\eta: F(G) \...
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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 ...
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What are some examples of groups each isomorphic to a subgroup of each other, although not themselves isomorphic?

Do there exist groups $G$ and $H$ for which $G$ is isomorphic to a subgroup of $H$ and $H$ is isomorphic to a subgroup of $G$, but in fact $G$ is not isomorphic to $H$? I know that $G = F_2$ and $H = ...
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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 ...
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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: ...
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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 ...
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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 ...
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1answer
88 views

Free groups. Help me to understand this theorem.

This is the Theorem 2.9 from book Lyndon, Schupp "Combinatorial group theory". Let $F$ - is free group, $G$ - is subgroup of $F$, $\forall g\in G$ define $G_g=Gp(\{h: h<g, h\in G\})$. So $A=\{g:g\...
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1answer
39 views

finite index subgroups in free group non-trivial intersection with each of the non-trivial subgroups of the free group.

I was reading a paper and find this statement in the abstract, "If $H$ has finite index in $F_m$, then $H$ has non-trivial intersection with each of the non-trivial subgroups of $F_m$" where $F_m$ is ...
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1answer
45 views

Proving Neilsen-Schreier Theorem using “only free groups act freely on a tree”

This is exercise II.9.16 from Aluffi's Algebra: Chapter 0. Before tackling this theorem, I have proved (rather loosely because I don't know much about graph theory) that the Cayley diagram of a free ...
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Representatives in the quotient of a lower central series of a free group.

Let $F$ be a free group of letters $\{x_1,...,x_m\}$ and define $$F_1:=F$$ $$F_k:=[F_{k-1},F]$$ According to this question, $F_2/F_3$ and $F_3/F_4$ are generated by representatives given by basic ...
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Why is the free group on two generators not a subgroup of $G$?

Let $G$ be generated by the elements $g$ and $\{e_i\}$ for $i\in\mathbb{Z}$, having the relation $ge_ig^{-1}=e_{i+1}$. It seems to me like there is a subgroup $\langle e_i,e_{i+1}\rangle$ for example,...
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1answer
122 views

Group freely generated by monoid

There are several ways to define the group freely generated by a monoid, all of which (necessarily) produce isomorphic groups. One way starts with a presentation of the monoid, and simply reinterprets ...
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1answer
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On free abelian groups

I'm learning about the concept of a free abelian group. First question: nowhere it is stated that these groups cannot be finite, but the definition seems to imply it. Is this true? Second question: ...
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1answer
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Any subgroup of free abelian group $\mathbb{Z}^r$ of index $n$ contains subgroup $n\mathbb{Z}^r$.

Any subgroup of index $n$ of free abelian group $G= \mathbb{Z}^r$ contains subgroup $n\mathbb{Z}^r$. My attempt to prove this is as follows: If I have any subgroup $F$ of index $n$ of G, then ...
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1answer
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Injective homomorphisms between group presentations

I'm studying the proof that $S_n = \langle s_1,...,s_{n - 1} \mid s^2_i = 1, (s_is_{i + 1})^3, s_is_j = s_js_i$ for $|i - j| > 1 \rangle$. The key part of the proof is that the subgroup of $\Gamma_{...
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1answer
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 ...
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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 ...
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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 ...
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1answer
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 $F$ ...
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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 ...
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1answer
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. ...
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89 views

How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?

Let $D_{2\cdot 8}$ be given by the group presentation $\langle x,y\mid xy = yx^{-1} , y^2 = e, x^8 = e\rangle$. Let $G = F_{\{x,y\}}$ be the free group on two generators and $N = \langle\{xyx^{-1}y,y^...
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Motivation for definition of free group?

Let $S$ be a set and $F_S$ be the equivalence classes of all words that can be built from members of $S$. Then $F_S$ is called the free group over $S$. I don't understand the motivation for this ...
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1answer
80 views

Constructing infinitely many automorphisms of a free group on two generators to itself

I want to use the universal property of the free group on two generators $F = F(\{a, b\})$ to construct infinitely many automorphisms of $F$, with the restriction that are not they are not inner ...
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15 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 $b_4(n)$? ...
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1answer
76 views

Free groups, generators and group homomorphisms

Let $F_2$ be the free group with generators $x_1$,$x_2$, and let $F_3$ be the free group with generators $y_1$,$y_2$,$y_3$. We define a group homomorphism $\phi:F_3\rightarrow F_2$ by $\phi(y_1):=...
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1answer
31 views

A Normal subgroup of a free group generated via a fixed integer

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: ...
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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 $F$ ...
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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 ...
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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 $(\...
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3answers
141 views

On the confusing definition of free group in Munkres

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 ...

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