# 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 ...
1 vote
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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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### 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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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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### 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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### 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 ...
### $F_X \cong F_Y \Rightarrow |X| = |Y|$ 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 = ...