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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Can the following proof calculus show that any finitely presented free group is free?
If a finitely presented group is free, will it always have a proof in the proof calculus outlined in this question that it is free?
I recently saw this question.
I tried to show that the group was ...
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Pushout of Z isomorphic to Z
$\mathbb{Z}\xleftarrow{\text{id}}\mathbb{Z}\xrightarrow{\text{x2}}\mathbb{Z}$
How do we prove this group pushout is isomorphic to $\mathbb{Z}$?-
I know we can take $\langle x|\rangle$ as a ...
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Presentation of Product Group
Here is the question I have been working on:
If $G_1 = \langle X_1 : R_1\rangle$ and $G_2 = \langle X_2 : R_2\rangle$, supply a presentation for $G_1 \times G_2$.
Deduce that, if $G_1$ and $G_2$ are ...
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Ping Pong Lemma
In the book "Groups, Graphs and Trees" by John Meier, Lemma 3.10 is the Ping Pong Lemma. He uses different assumptions than for example Wikipedia. Namely, he states that
Let $G$ be a group ...
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How do we generate the loop $ba$ from the loops $a^2,b^2$ and $ab\ $?
In the second diagram a $2$-sheeted connected covering of the figure eight has been described. The image of the fundamental group of the covering space has the generators $a^2, b^2$ and $ab$ as ...
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Has anyone studied differential equations on $\mathbb{Z}F_n$ defined using Fox derivatives?
I am looking for a reference, if it exists, to the study of differential equations defined using Fox derivatives over the group ring, say, of a free group. Is this a topic which has been studied ...
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Is group defined by action on $\mathbb{R}$ a free group?
Assume that I have two bijective functions $f,h:\mathbb{R}\to\mathbb{R}$ such that $f(h(x))\neq h(f(x))$ as well as $f(x)\neq x \neq h(x)$ for all $x\in\mathbb{R}$. Additionally, I consider the group $...
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Show a free group has no relations directly from the universal property
The free group is often defined by its universal property. A group $F$ is said to be free on a subset $S$ with inclusion map $\iota : S \rightarrow F$ if for every group $G$ and set map $\phi:S \...
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The fundamental group of closed orientable surface of genus 2 contains a free group on two generators
Let $S$ be the closed orientable surface of genus $2$. It is well known that its fundamental group is given by $$ \pi_1(S)=\langle a,b,c,d:[a,b][c,d]=1\rangle.$$
How can we show that this group has a ...
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Proving that 2 formulations of the Universal Property of Free Groups are equivalent
The present question is inspired by an answer to this question of mine on applications of Category Theory in Abstract Algebra.
One of the answers stated that the universal property of free groups ...
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Epimorphism between free groups that inject on a finite subset
I asked a question on MathOverflow (https://mathoverflow.net/q/454012/513011) where the following lemma appeared:
Folklore lemma: Let $S$ be a finite subset of the free group $F_n$ of finite rank $n$....
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Construction of left adjoint
A very standard example of left adjoints would be the functor $F:Sets\to Grps$ that maps sets to their free groups is a left adjoint to $G:Grps \to Sets$ which forgets the group structure. My question ...
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The group generated by two homeomorphism $f,g$ is free group?
In my manuscript, we assume that $F_2=\langle a, b\rangle$ is free group.
Also, we assume that $\varphi:F_2\times X\to X$ is generated by two homeomorphism $\varphi_a=f$ and $\varphi_b=g$.
Referee ask ...
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Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(\mathbb F_d,\...
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Epimorphism from $F_n$ to $F_{n-1}$?
$F_n$ is the free group of rank $n$. I know that it is possible to construct an Epimorphism $f: F_n \to F_{n-1}$. How is this done? (I also want to consider the case $F_2 \to \mathbb{Z}$ but maybe ...
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Three questions about Wikipedia's definition of Van Kampen's theorem for fundamental groups
I'm studying Algebraic Topology off of Hatcher and (unfortunately as usual) I find his definition and explanation of Van Kampen's theorem to be carelessly written and hard to follow. I happen to know ...
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Normalizer of Factors in an Amalgamated Free Product
I am reading through this proof that if $H$ is a non-trivial group, then the normalizer of $H$ in the free product $G:=H \ast K$ is equals $H$ (i.e., is trivial). Most of the proof seems to generalize ...
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Rank of subgroups of free groups
I am familiar with Schreier's theorem which states that any subgroup H of a free group G is free, as well as with the formula
$$
\operatorname{rank} H - 1 = |G:H|(\operatorname{rank} G - 1)
$$
I have ...
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Mapping class group of surfaces, free products, and trees
Let $ \Sigma $ be a surface, possibly with boundary. Let $ MCG(\Sigma) $ denote the mapping class group. Is it true that $ MCG(\Sigma) $ has a quotient which is a nontrivial free product $ A \ast B $ ...
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Are any quotients of braid groups non-trivial free products?
The braid group $ B_1 $ is trivial and the braid group $ B_2 $ is isomorphic to $ \mathbb{Z} $.
The braid group $B_3$ has the property that its central quotient (i.e., $B_3 / Z(B_3)$) is isomorphic to ...
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Free product of finite groups that is outside graph theory
The free product of finite groups $ A * B $ naturally acts on a biregular graph see Free Product of two finite groups. This seems like one of the only places that free products of finite groups appear ...
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A group generated by two elements with uncountably many normal subgroups
Proposition. There exists a group generated by two elements with uncountably many normal subgroups.
The constructive proof below appears in Geometric Group Theory: An Introduction by Clara Löh (...
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If $a^{p_1} b^{q_1}\ldots a^{p_n} b^{q_n} = e$ then $S = \{a,b\}$ is not a free generating set of $G = \langle S \rangle$
Let $S = \{a,b\}$ be a generating set for a group $G$. If a non-trivial word in $a, b \in S$ equals the identity $e$ of $G$, i.e.,
$$a^{p_1} b^{q_1} a^{p_2} b^{q_2} \ldots a^{p_n} b^{q_n} = e$$
for ...
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Computing $\mathrm{Fix}(\phi)$ for autormophisms $\phi$ of free groups
Let $F_A$ be the free group generated by the finite set $A$ and let $\phi\colon F_A \to F_A$ be a group-automorphism. It is known [1] that
$$ \mathrm{Fix}(\phi) = \{g \in F_A : \phi(g) = g\} $$
is (...
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$\prod_p \mathbb{Z}/p\mathbb{Z}$ is not the direct sum of $\bigoplus_p \mathbb{Z}/p\mathbb{Z}$ and a torsion-free subgroup
While I was reading "Abelian Groups" by Fuchs $(2015)$, I encountered Example $1.2$ in the chapter on Mixed Groups, which stated the following:
Let $p_1,p_2,\dots,p_n,\dots$ denote different ...
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Questions to limit groups (over free groups)
My questions refer to the following article (both refer to page 27):
https://arxiv.org/pdf/2002.10278.pdf
In the article we find the statement that for a non-abelian limit group $L$ we always find a ...
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Understanding why this "obvious" approach to Schreier-Nielsen's theorem doesn't work [duplicate]
While talking about various proofs of Nielsen-Schreier Theorem, someone asked a question and I couldn't come up with a satisfying answer on the spot. I was discussing how there's a nice proof using ...
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Is $\Bbb Z^3$ a one-relator group?
I understand that:
$\Bbb Z^0 = \langle a \mid a \rangle$
$\Bbb Z^1 = \langle a, b \mid b \rangle$
$\Bbb Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$
but is it possible for $\Bbb Z^3$ to be ...
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A subgroup $H$ of an abelian group $G$ is a direct summand iff $G/H$ is free?
Let $G$ be an abelian group and $H\le G$. Then TFAE:
$G/H$ is free.
$H$ is a direct summand of $G$.
This is how I tried to prove this theorem:
If $G/H$ is free, then $G/H$ has a basis. Let's denote ...
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Let $H ≤ F_A$ be finitely generated and $g \in F_A$. Show that it is decidable if $xgx^{-1} \in H$ for some $x \in F_A$
By the generalized word theorem, we know that if we have $H ≤ F_A$ finitely generated and $g \in F_A$, then $g \in H$ iff $\overline{g}$ is accepted in $S_H$ (Stalling's Automata of H).
Then, $xgx^{-1}...
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Recovering an element of a free group from its projections
Assume you have an unknown word on an alphabet with at least three letters, and you know all the words obtained by erasing each copy of some letter. Then, you can find the first letter of the original ...
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Question about group of rank Aleph-Null.
Definition : A group $G$ is said to have a finite rank $r$ if every finitely generated subgroup can be generated by $r$ elements, and $r$ is the least positive integer with this property. If no such ...
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Why doesn't the infinite dihedral group contain a free subgroup of rank 2?
Our professor just told us that $D_{\infty}$ is "too small" whatever that means. Can someone prove this statement and give some reason as to why this statement holds true but doesn't for ...
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a doubt on free group in Dummit&Foote's Abstract Algebra
I have a doubt about free group in Dummit's Abstract Algebra on page220 :
More generally, suppose $G$ is presented by, say, generators $a, b$ with relations $r_1 , . . . , r_k$ . If $a', b'$ are any ...
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Definition of hyperbolic elements and axes in a (limit) group
I am writing on my master thesis at the moment and it is based on the following article of Fujiwara and Sela: https://arxiv.org/abs/2002.10278
On page 7 they use the terms "hyperbolic element&...
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For $1 \le i \lt j \le n$ and $d=j-1$, prove $S_n = \langle\{i\;\; \overline {i+d} : 1 \le i \le n\}\rangle$ $\iff$ $gcd(d,n) = 1$
So in the first part of the question I had to prove that $gcd(d,n)=1 \implies S_n = \langle\{(i\;\; \overline {i+d}) : 1 \le i \le n\}\rangle$ whilst $\forall a \in \mathbb {Z}$,
we mark $\overline {a}...
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Normal subgroup of fundamental group of Klein Bottle
Let $K$ the Klein Bottle and $\pi_1(K) = \langle a,b \mid b a b a^{-1} \rangle $ be the fundamental group of the Klein bottle. Observe that $\langle b \rangle $ is a normal subgroup of $\pi_1(K)$, ...
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Minimal generating set of $p_*(\pi_1(E,e))$
Consider the following degree $4$ non-normal covering space of $S_1\lor S^1$ I drew:
Here, $a$ and $b$ denote the edges which map onto the first and second circle in $S^1\lor S^1$ respectively. I ...
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Question about functoriality of free group.
This is an example from Riehl's category theory in context specifically 1.3.2.ix. She writes the following:
Example: There is a functor $F:\mathbf{Set} \rightarrow \mathbf{Group}$ that sends a set $X$...
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Rank of free groups
In Johnson's 'Topics in the Theory of Group Presentations', one can find this theorem after the definition of free groups using the universal property.
Theorem. Free groups of different ranks are not ...
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Injective map needed in definition of free groups
I have the following definition:
Let $X$ be a set. The free group genereted by $X$ is a group $F$, if there exists an injection $i:X\to F$ s.t. for all Groups $G$ and (not neccesarly injectiv) ...
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a typo about free groups in Dummit's Abstract Algebra
I am not sure that if there is a typo in Dummit's Abstract Algebra on page219 : Let $S=G$ and the map $\pi:F(S)\to G$ is the homomorphism extending the identity map of $S$ . the first paragragh writes ...
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Let $\Bbb Z*\Bbb Z=\langle a,b\rangle$ and $N=\{waba^{-1}b^{-1}w^{-1}:w\in\Bbb Z*\Bbb Z\}.$ Prove $\langle a,b\rangle/N$ is abelian
Let $\mathbb{Z} * \mathbb{Z} = \langle a,b \rangle$ and $$N = \left\{w a b a^{-1} b^{-1}w^{-1}: w\in \mathbb{Z} * \mathbb{Z} \right\}$$ the smallest normal subgroup that contains $\left\{ a b a^{-1} ...
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Index of an infinite cyclic subgroup of a free group of infinite rank is infinite
Question: Let $G$ be free of rank at least $2$, and let $H$ be an
infinite cyclic subgroup of $G$. Is it true that the index $[G: H]$ of
$H$ in $G$ is infinite?
My thought: I know the case when the ...
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Given a subgroup of a free group, find the associated covering space.
Let $R_2$ the rose with $2$ petals, that is the wedge of $S^1$ with itself. We know its fundamental group is the free group with two elements, $\pi_1(R_2)=F_2=\langle a,b\rangle$. Now given some ...
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In the free pro-C constructions is enough to verify the universal property for C-groups
The book Profinite Groups of Ribes-Zalesskii shows the existence of free constructions using a similar ideia in all of them (free pro-C groups, free pro-C products, etc). In these proofs the author ...
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Computation of Amalgamated Product $\mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6.$
I'm trying to compute a amalgamated product $\mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6$.
Let $\mathbb{Z}_4= \langle a\mid a^4 =1\rangle$ and $\mathbb{Z}_6 = \langle b\mid b^6 =1\rangle $, be a ...
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Construction of Free abelian groups on Massey Book
I am reading the book of Massey of Algebraic topology, and I am having trouble to understand this construction.
Let $ S = \left\{ x_i : i\in I \right\}$. For each index $i$, let $S_i$ denote the ...
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$F_X \cong F_Y \Rightarrow |X| = |Y|$ where is the mistake in this proof
Statement.
Let $F_X, F_Y$ free groups over $X,Y$ respectively. Suppose there is an isomorphism $\phi: F_X \cong F_Y$. Then $|X|= |Y|$.
My proof. Let $x \in F_X$ be a word of length one, this is, $x = ...
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Why must reduced words in a free group be component-wise equal to be equal?
Edit: alright, I guess I will provide the complete quote of the page, and will not leave out a single detail so that my question is comprehensive.
I am reading Abstract Algebra: 3rd Edition by Dummit ...
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