# Tagged Questions

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### Rank of a free group

I am trying to know whether the following result is true. Let $F$ be a free group with a basis $X$, and let $X'=\{xF':x\in X\}$, where $F'$ is the commutator subgroup of $F$. Then, $|X|=|X'|$. ...
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### Semidirect products as (amalgamated) free product

It is well known that $\mathbb{Z}\rtimes \mathbb{Z}_2$ is free product $\mathbb{Z}_2 \star \mathbb{Z}_2$. Are there more examples of these kind?
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### Pairs of $2\times 2$ matrices generating free groups.

The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, ...
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### Linear representation of a free group

I need to prove the following: "Prove that the free group of rank 2 is linear." So to my best understanding, and please correct me if I'm wrong: I actually need to show a homomorphism from the free ...
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### The action of free group on line

Let $G$ be a free group, if the action of $G$ on $\mathbb{R}$ is free, does it imply that $G$ is abelian?
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### the center of amalgamated product of free groups

Let $G_1$, $G_2$, $H$ be free groups, $K=G_1*_H G_2$ is the amalgamated product of free groups, then is center of $K$ trivial? Thanks in advance.
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### Finitely generated groups without the minimal condition on subgroups

Definition A group $G$ is said to satisfies the minimal condition on subgroups iff every descending chain of subgroups stops after a finite number of steps. I know Novikon and Adjan proved that the ...
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### An old problem in Group Theory

Definition A group $G$ is said to satisfies the minimal condition on subgroups iff every descending chain of subgroups stops after a finite number of steps. I know this was an unsolved problem in ...
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### Show that the group $\langle a, b, c | ab = bac \rangle$ is free

I'm trying to use Tietze transformations tranforms it to the group $\langle a, b, c \rangle$ - is this the right thing to do?
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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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### 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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### 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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### 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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### 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 ...
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 ...