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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Confusion related to definitions involving free groups

From Wikipedia ...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different ...
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Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
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Showing a Mapping Between $\left\langle a,b \mid abab^{-1}\right\rangle$ and $\left\langle c,d \mid c^2 d^2 \right\rangle$ is Surjective

Hypothesis: Let $$ G \cong \left\langle a,b \mid abab^{-1}\right\rangle $$ $$ H \cong \left\langle c,d \mid c^2 d^2 \right\rangle $$ Let the function $f$ be defined as follows. First let $f(a) ...
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The Relationship Between Generators, Relations and Group Homomorphisms

Let $G = \left\langle g_1, \ldots , g_n \mid R_1\right\rangle$ and $H = \left\langle h_1, \ldots , h_n \mid R_2 \right\rangle$. Suppose there's a mapping $f$ s.t. $f(g_i) = h_i$. Can we then merely ...
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Verifying $G*H$ Has Trivial Center and Elements of Infinite Order

Hypothesis: Let $G \ne H$ denote two non-trivial groups. Goal: Show that $G * H$ has a trivial center (hence is non-abelian) and contains an element of infinite order. Is my attempted proof below ...
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Verifying $G*H \cong G' * H' \implies |G| = |G'|$ or $|G| = |H'|$ (All Groups Cyclic)

Hypothesis: Let $G$, $H$, $G'$, and $H'$ be cyclic groups of orders $m$, $n$, $m'$, and $n'$ respectively. Goal: Show that if $G * H$ is isomorphic to $G' * H'$ then $m = m'$ and $n=n'$ or else $m ...
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55 views

Notion of group generators

I'm asking myself what the meaning of a statement like the following is: Let $G$ be a group and $T_i \subseteq G$ be a family of subgroups of $G$ indexed by a possibly infinite set $I$. Now let ...
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39 views

A multiplicative subgroup of rational numbers

Let $\Bbb Q^+$ be the set of positive rational numbers and $K=\{\sqrt {t^2 +s^2}:t, s\in \Bbb Q^+\}$. It is easy to see that $K$ forms a multiplicative group since $$ ...
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Isomorphism between free pro-$p$-groups induced by isomorphism on abelianizations

Let $G$, $H$ be free pro-$p$-groups, where $p$ denotes a rational prime. I want to show that a canonical homomorphism $$G \to H$$ is in fact an isomorphism. The next step is a reduction step to ...
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Defect of a quasimorphism

Suppose we have a free group $F_2$ on 2 generators (say, $g_1$ and $g_2$) and and element $f=f_1 ... f_k \in F_2$ such that each $f_i$ can be one of $\{g_1,g_1^{-1},g_2,g_2^{-1}\}$ and of course ...
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Free group and normal subgroups with corresponding schreier representatives.

Let $\mathbb{F}$ be a free group on $a$ and $b$. Let $N$ be the normal subgroup of $\mathbb{F}$ generated by $a^2,b^3$ and $(ab)^2$. Similarly $H$ is the normal subgroup generated by $a^2,b^3$ and ...
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1answer
56 views

Freely generated groups and homomorphisms

Let $G$ be a group and $\{x_1,x_2,\ldots,x_n\}$ a set of its elements, such that for any group $F$ and any set $\{y_1,y_2,\ldots,y_n\}$ of elements of $F$ there is one and only one homomorphism $h: G ...
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67 views

Reducing word algorithm on the free group construction

Let $X$ be a non-empty set. There exists sets $X'$ and $\varepsilon$ such that $\varepsilon \notin X \cup X'$, $X' \cap X = \varnothing$ and $|X|=|X'|$, ie, $X$ and $X'$ have the same cardinality. ...
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Basis for singular chains group

The singular chain group $S_p(X)$ is defined as the free abelian group generated by continuous functions $T \in C( \Delta_p , X)$. What I understand this means is that we define $T' \in S_p(X)$ as ...
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What can we say about the kernel of $\phi: F_n \rightarrow S_k$

Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the ...
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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 $X$?
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Why isn't $\langle a ; a^2 \rangle$ (or $\langle a;a^3, a^7\rangle$) a presentation of $C_4$?

I've just read the first few pages of Combinatorial Group Theory by Magnus, Karrass, and Solitar, and based on their definitions there, and more specifically, the reasoning given in the hint to ...
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1answer
92 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. Can we say for sure, whether this presentation defines a nontrivial group?
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Subgroup of a (free) group.

Let $F$ be a group, generated by $x_1,...,x_m$ and $H$ be its subgroup such that $|F:H|=n < \infty$. How to prove that $H$ can be generated by $n(m-1)+1$ elements?
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Group presentations - again

My question is about finding presentations for finite groups. It's along similar lines to my earlier question -- but is subtly different! The earlier question is here Group presentations Let's take ...
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1answer
62 views

Does every group have a representation?

For any set $A$, we can give it an group structure and make it a free group. For example: $$\mathbb Z=<a;aa^{-1}=1>$$ Further more, we can introduce some relation on it: $$\mathbb ...
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How many normal subgroups is in a free group of rank > 1

How many normal subgroups is in a free group of rank $k>1$, if the quotient group (of the normal subgroups) isomorphic $S_3$?
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Two free basis of a free abelian group

We have a free abelian group $A(X)$, where $X$ is its free basis, and let $Y$ another free basis for $A(X)$. We know that every $g\in A(X)$ can be expressed as $g=a_1x_1+...+a_nx_n$ where the $a_i$'s ...
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Free groups contain “larger” free groups

First of all, I have read this similar question and am satisfied that the answers there prove the result I am interested in. That being said, I'm more interested in this particular approach than in ...
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348 views

Prove elements generate a free group

How does one show that the elements $x^2$, $y^2$, and $xy$ have no nontrivial relations among them in the free group generated by $\{x,y\}$? This would prove that the free group $F_2$ has a subgroup ...
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A group presentation for $\mathbb{Z}_2\times \mathbb{Z}_2$

I know that the only groups of order 4 are $\mathbb{Z}_2\times \mathbb{Z}_2$ and $\mathbb{Z}_4$ up to isomorphisms. And I also know that the group presentation of $\mathbb{Z}_4$ is $\left ( a:a^4=1 ...
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1answer
61 views

locating a square root element with square in a free subgroup

Let $G$ be a group (not necessarily countable discrete) which contains a free subgroup $F=F_2=\langle a, b\rangle$, denote $H=\langle a, b^2\rangle$, and assume that the centralizer of $F$ inside of ...
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1answer
60 views

Alternative approach to constructing the free group

Let $A$ be a set. We wish to construct the free group $F(A)$. It seems that this (invariably?) starts out like this: Let $A'$ be a copy of $A$, and let $\mathscr A=A\cup A'$. Let $\mathscr L$ ...
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Let A be a finitely generated abelian group. Show that Hom(A,Z) is a free abelian group.

My question is Let $A$ be a finitely generated abelian group. The structure theorem says that $A$ is isomorphic to $F \times T$, where $F$ is isomorphic $\mathbb Z^m$, some $m \geq 0$, and $T $ is ...
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1answer
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Finding generators for a non-normal group

How would I find the generators for a non-normal index 3-subgroup of the free group $\langle a,b| - \rangle$ ?. I know that a finitely generated free group can be realised as the fundamental group of ...
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1answer
52 views

Connected Covering Space of a Bouquet of 3 Circles

Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. What is $\pi_{1}(\hat{X})$ (It is a free group of what rank?). Any hints?
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Bouquet of Two Circles and Based covers

I need help with the following: Let $B$ denote the bouquet of two circles labeled by $a,b$. We regard $\pi_{1}(B)\cong \langle a,b | -\rangle $ where the basepoint of $B$ is its vertex, and the two ...
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Automorphism of the free group

Let $\mathbb{F}_2$ be the free group of rank $2$ with generators $a$ and $b$. I would like to build an automorphism $\varphi$ of $\mathbb{F}_2$ such that : 1) $\varphi([a,b]) = [a,b]$ 2) ...
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Could the concept of “finite free groups” be possible?

Is it possible to define "finite free groups" ? could that make it easier to deal with group presentations ?
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The quotient of a free group, of rank $n$, and its commutator subgroup is isomorphic to $\mathbb{Z}^n$

Let $F$ be a free group of rank $n$. Let $G$ be the commutator subgroup of $F$. I need prove that $F/G\cong\mathbb{Z}^n$. I have tried with the isomorphism theorem: With $\varphi$, I send the $x_i$ ...
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Free subgroups of piecewise linear homeomorphisms of the circle

For convenience, I recall the definition of piecewise linear homeomorphism: A homeomorphism $f$ of the real line $\mathbb{R}$ is called piecewise linear if there is an increasing sequence of real ...
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subsets of a generating set of a free group

Is it true that, given any free group $F$ and any set $S\subseteq F$ which generates $F$, there is a subset $T\subseteq S$ which is a free generating set for $F$? It would be great if you could give ...
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Specific basis of a subgroup of a free abelian group

I'm looking for clarification on Fraleigh's "A First Course in Abstract Algebra" Theorem 38.11. It states: "Let $G$ be a nonzero free abelian group of finite rank $n$, and let $K$ be a nonzero group ...
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96 views

Groupe generated by elements with a relation

Could someone show me how to prove that the group generated by $x,y,z$ with the single relation $yxyz^{-2}=1$ is actually a free group.
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Conjugacy class of commutators of generators for a free group.

Let $F$ be a free group generated by two elements. Let $\{a,b\}$ and $\{c,d\}$ are two different generating set. Q:Prove that $[a,b]$ is either conjugate to $[c,d]$ or its inverse. Here ...
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software to decide whether a 2-generator subgroup of PSL(2,R) is discrete/free

Gilman developed an algorithm with polynomial complexity that, given two elements in PSL(2,R), decides whether the group they generate is free/discrete or not. I was wondering whether anybody ever ...
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free object isomorphisms

In group category if $F_1$ be a free object on $X_1$ and $F_2$ is free object on $X_2$ and $F_1$ isomorphic to $F_2$ prove that |$X_1$|=|$X_2$| whats the relationship between isomorphisms of free ...
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free groups , a question of listing elements and drawing multiplication table

I'm requested to list the elements and draw the multiplication table for the group $\langle a, b : |a| = 2 = |b|\rangle$ without any more details. But hence this group is infinite isn't it ? while ...
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Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
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Group with presentation $<x,y \ | \ x^2, y^2>$ is generated by $2$ elements of order $2$

Could you tell me how to prove that a group with presentation $<x,y \ | \ x^2, y^2>$ is generated by $2$ elements of order $2$? I know it's infinite, because we will have infinitely many ...
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1answer
110 views

free groups: no relations between generators

Consider the free group $\langle x,y\rangle$. I'd like to show that $x^2,y^2,xy$ have no relations between them, without the theorem that a subgroup of a free group is free, without knowledge about ...
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72 views

All bases for a finitely-generated abelian group have the same cardinality.

I want to understand more about this proof from Lang's Algebra: Let $B$ be a subgroup of a free abelian group $A$ with basis $(x_i)_{i=1...n}$. It has already been shown that $B$ has a basis of ...
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Nilpotency of the $n$-fold free product of cyclic groups of order 2

Beforehand: I am not particularly algebraically educated and, especially, I do not have much background as far as free products of groups are concerned. So, it might well be that my question seems ...
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Automorphisms of free groups

Suppose $U$ is a subgroup of finite index in the free group on $k$ generators $F_k$. Suppose $\sigma$ is an automorphism of $F_k$ such that $\sigma|_U = \text{id}$, then must $\sigma = \text{id}$?
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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 ...