# 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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### Proving that free group actions are faithful.

Assume group $G$ acts on set $X$. My course notes say this: Free action is a special case of faithful action. To have faithful action it suffices that any $g \neq e$ in $G$ moves something (but not ...
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### Good introduction to free groups and free products

In my undergraduate research project, I am going to study a paper on free products in division rings. To do this, however, I, of course, need to learn about free groups and free products. Right now, ...
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### Quotient of a free group on a set

Suppose I have a free group $F_S$ on a set $S,$ with a nonempty proper subset $T.$ If $N$ is the smallest normal subgroup of $F_S$ containing $T$, I would like to show that $F_S/N$ is also free. There'...
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### When does a RAAG act on a Tree?

Show that a RAAG acts on a tree of valence 4, acting transitively on vertices, if at least one pair of vertices is not joined by an edge. I'm trying first to prove that every such RAAG $G$ acts on a ...
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### Free monoid on indexed sets and arbitrary operations

In the proofwiki, there is this definition: The free commutative monoid on an indexed set X=⟨Xj:j∈J⟩ is the set M of all monomials under the standard multiplication. That is, it is the set M of all ...
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### Whether a group containing a free group is also a free group

Suppose $G$ is a group generated by two elements $s,t$. Suppose $H$ is a subgroup of $G$ such that $H= \langle s^k,t^k \rangle$ is a free group where $k$ is some integer not equal to $1$. Does it ...
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### Show that the Abelianized group $G=\langle t_1,\dots,t_n;t_{1}^{2}t_{2}^{2}\dots t_{n}^{2}\rangle$ is $\Bbb{Z}^{n-1}\oplus\Bbb{Z}/2\Bbb{Z}.$ [closed]

Show that the Abelianized group $G=\langle t_1,\dots,t_n; t_{1}^{2}t_{2}^{2}\dots t_{n}^{2}\rangle$ is $\mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z}$. I don't have any idea how to solve this. In ...
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### Example of homomorphism $\underset{\alpha \in A}{\ast} \pi_1(U_\alpha) \to \pi_1(X)$, where $U_\alpha \ni x_0$ are an open cover of $(X,x_0)$.

I’m reading a passage leading up to the Van Kampen’s theorem, and I have trouble understanding the settings. Here’s the materials. Let us consider a pointed space $\left(X, x_{0}\right)$ with an open ...
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### Why does no non-trivial reduced words in $H_1 \backslash \{e\} \sqcup H_2 \backslash \{e\}$ imply that $\left< H_1, H_2 \right> = H_1 \ast H_2$?

I am currently studying combinatorial group theory and am trying to prove the ping pong lemma. Many of the proofs I have come across seem to use a result of the following flavour. Let $G$ be a group ...
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### Free group $F(a,b)$ of rank 2 and free subgroup of rank $\infty$

Q: $F(a,b)$ is free group of rank 2, give an example (and proof it!) of free subgroup with infinity rank. My attempt: I was trying with $G=[F(a,b), F(a,b)]$ and $G=\langle {a^nb^n : n\in N}\rangle$ ...
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### Show that F2 has an infinite index free subgroup of rank 2 [duplicate]

I am trying to show that F2 (the free group on 2 elements) has an infinite index, free subgroup of rank 2. As it has to be a free subgroup of rank 2 I'm guessing I need to find 2 elements in F2 that ...
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### Retract of $F_2$ onto $[F_2,F_2]$

I'm trying to show that the commutator subgroup $[F_2,F_2]$ is not a retract of $F_2$. I was trying to do the proof by contradiction so I assume there is a retract $f: F_2 \to [F_2,F_2]$, then I know ...
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Someone could help me with this proof contained in "A course in the theory of Groups" by Derek Robinson? Let $F$ be a free group on a subset $X$. If $Y$ is a nonempty subset of $X$, prove ...
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### Free Groupoid over a finite directed chain

I'm trying to understand a free groupoid on a directed chain of n vertices. I know the objects will be the vertices of our graph and the morphisms will be concatenations of the edges in the graph and ...
Let $G = (\mathbb Z/2\mathbb Z)^{\ast m}$ be a free product of some groups of order $2$. Let $\alpha_1,\ldots,\alpha_m$ be the generators. Can I find a free, nonabelian subgroup of $G$ that has no ...
P. Aluffi's "Algebra: Chapter $\it 0$", exercise II.$5.8$. Still more generally, prove that $F(A\amalg B)=F(A)*F(B)$ and that $F^{ab}(A\amalg B)=F^{ab}(A)\oplus F^{ab}(B)$ for all sets $A,B$...