1
vote
2answers
57 views

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 ...
3
votes
3answers
60 views

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) ...
0
votes
1answer
18 views

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 ...
0
votes
0answers
28 views

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 ...
1
vote
0answers
25 views

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 ...
0
votes
1answer
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 ...
0
votes
0answers
18 views

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 ...
1
vote
2answers
48 views

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 ...
2
votes
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 ...
2
votes
2answers
48 views

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 ...
5
votes
2answers
102 views

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 ...
12
votes
5answers
634 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 $X$?
-2
votes
1answer
104 views

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 ...
5
votes
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?
1
vote
0answers
43 views

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?
1
vote
2answers
91 views

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 ...
1
vote
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 ...
2
votes
1answer
91 views

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 $S_3$?
2
votes
1answer
88 views

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 ...
11
votes
5answers
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 ...
2
votes
3answers
91 views

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 ...
1
vote
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 ...
2
votes
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 $\mathscr L$ ...
5
votes
1answer
65 views

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) ...
11
votes
3answers
389 views

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 ?
0
votes
1answer
56 views

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 ...
0
votes
1answer
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.
1
vote
0answers
88 views

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 ...
0
votes
1answer
48 views

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 ...
3
votes
5answers
90 views

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?
1
vote
0answers
89 views

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 ...
1
vote
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 ...
0
votes
2answers
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 ...
2
votes
1answer
59 views

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 ...
4
votes
1answer
41 views

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 $\sigma = \text{id}$?
3
votes
1answer
121 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
vote
1answer
39 views

Induced injections between free groups

Let $A$ and $B$ be non-empty sets with associated free groups $F(A),F(B)$. Given an injective function $f: A \to B$, is the induced homomorphism $\bar{f}: F(A)\to F(B)$ injective? Let $i_A: A \to ...
0
votes
1answer
112 views

Free abelian subgroup of index 2.

Let $G$ be a group with the following presentation $G=gp(x,y \mid x^2=y^2=1)$. I need to know, what further information about $G$ can be derived from knowing that $G$ has a free abelian subgroup of ...
3
votes
0answers
107 views

Free groups and their commutator subgroups.

I was trying to understand the example 3 of this paper of Baer. Take a free group $F$ of rank 2. Take the commutator group $F'$ (this is a free group having infinite countable rank). Then say $N$ the ...
1
vote
1answer
228 views

Existence of an automorphism of $F_2$ having some certain property

Let $F_2 = \langle a, b \rangle$, the free group of rank $2$. Is there an automorphism $\phi$ of $F_2$ such that $\phi(b) = b$, $\phi^2 = 1$ and $\phi(a) = wa^{-1}bw^{-1}$ for some $w \in F_2$? ...
2
votes
3answers
112 views

Matrices as generators of free group.

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing... In my words: The matrices $\begin{pmatrix}1 & 0 \\ 2 ...
4
votes
0answers
63 views

Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
4
votes
2answers
135 views

Is $\Bbb{6Z}$ a free group?

I'm trying to understand the concept of free groups , and from what I've learned so far , a group $G$ is called a free group , if there is a subset $S ⊂ G$ such that any element of G can be written ...
2
votes
1answer
83 views

Prove that f induces a unique homomorphism F/[F, F] → G and conclude that F/[F, F] ≅ Fab(A).

Let $F = F(A)$ be a free group, and let $f : A \rightarrow G$ be a set-function from the set $A$ to a commutative group $G$, and where $[F,F]$ is the subgroup of F generated by all elements of the ...
5
votes
1answer
275 views

Power of commutator formula

A few people remember a commutator formula of the form $[a,b]^n = (a^{-1} b^{-1})^n (ab)^n c$ where $c$ is a product of only a few commutators (say $n-1$) of them. Here $a,b$ are in a (free) group ...
0
votes
1answer
46 views

free groups question

I need help to solve this question If $f:G_1\to G_2$ and $g:H_1\to H_2$ are homomorphisms of groups, then there is a unique homomorphism $h:G_1*H_1\to G_2*H_2$ such that $h_{G_1}=f$ and ...
2
votes
5answers
319 views

Every nonidentity element in a free group $F$ has infinite order

I'm trying to prove that every nonidentity element in a free group $F$ has infinite order. I'm really new on free groups and I found this subject really strange I couldn't understand it very well yet, ...
3
votes
0answers
114 views

Combinatorial Group Theory / Lyndon & Schupp, Proposition 4.1.. Automorphisms of free groups

I have problem to understand the three final lines of the proof of the proposition 4.1 in page 23. The proposition is a generalization of the fact that $Aut(F_{n})$ is f.g. It says that if $F$ is a ...
6
votes
2answers
49 views

Free groups and derivative

Per definition a derivative on a group $G$ is a mapping $D:G\rightarrow\mathbb{Z}G$ such that $D(gh)=D(g)+gD(h)$. Now my question: uppose $G$ is a free group $F=F(X)$ with $X$ a finite set and suppose ...
5
votes
2answers
191 views

Lower central series of a free group

Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding ...