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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A question about commutators in free groups
Let $F$ be the free group on $X=\{ x_1,\dots, x_n\}$ for some $n\geq2$. Define the lower central series of $F$ inductively: $\gamma_1(F):= F$, $\gamma_{i+1}(F)=[\gamma_i(F),F]$ for $i\geq1$. Is it ...
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Condition on finitely generated subgroup of $GL_2(\mathbb{Q})$ to be free
I am considering finitely generated subgroup $G=\langle A,B,C\rangle$ of $GL_2(\mathbb{Q})$ such that $A,B,C$ all have the upper triangular form
$$A=\begin{pmatrix}a_1 & a_2 \\ 0 & a_3\end{...
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Virtual solvability of dense subgroups
Let $G$ be a (finitely generated) dense subgroup of $\mathsf{SL}(2;\mathbb{C})$. Is it possible that $G$ is virtually solvable?
In other words, by Tit's alternative, does being dense necessitate the ...
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Exercise on Generators and Relations from Michael Artin's book
The question is:
Let $\phi: G \mapsto G'$ be a surjective group homomorphism. Let $S$ be a subset of $G$ whose
image under $\phi$(S) generates $G$', and let $T$ be a set of generators of $\ker\phi$. ...
- 21
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Prove a surjective endomorphism $\phi$ of a 1-relator group $ ⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩ $ is not injective
Consider the infinite group $H$ with presentation
$$
⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩
$$
so that the relation is $a^{-1} b^2 a=b^3$.
The map
$$
a ↦ a\\b ↦ b^2
$$
induces a surjective homomorphism $ϕ:H\to ...
- 3,047
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How to show that the trivial group is the free group of the empty set (using universal property of free groups)?
Aluffi (in Algebra: Chapter 0) says that given a set $A$, the free group is a group $F(A)$ together with a set map $j_*:A\to F(A)$ st for any group $G$ and any set map $f:A\to G$, there is a unique ...
- 483
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$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point
Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
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Free group on $X$ means no relation in $X^{\pm}$ [closed]
I am reading Free Groups from the book ``Presentations of Groups" by D. L. Johnson.
The author says that the existence of $\theta'$ means there is no relation in $X^{\pm}$. He gives the argument ...
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Finding free subgroup $F_2$ in the free product $\frac{\mathbb{Z}}{5\mathbb{Z}} * \frac{\mathbb{Z}}{6\mathbb{Z}}$
Is there any free group isomorphic to $F_2$ contained in the free product group $\frac{\mathbb{Z}}{5 \mathbb{Z}}* \frac{\mathbb{Z}}{6 \mathbb{Z}}?$
Let $\frac{\mathbb{Z}}{5\mathbb{Z}}= \langle a \mid ...
- 435
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Burnside groups with GAP system [closed]
My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP.
The obvious representation using relations (see example for ...
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Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same?
I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student ...
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If there is a bijection from a subset $S$ of a group $G$ onto $X$ then $F(X)$ isomorphic to $\langle S \rangle$, Where $F(X)$ free group on X
Let $\phi: G \to F(X)$ be a group homomorphism suppose that $\phi$ maps a subset $S$ of $G$ bijectively onto $X$. Then $F(X) $ is isomorphic to $\langle S\rangle$, where $F(X)$ free group with basis $...
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extension condition for free abelian groups
if $G$ is a free abelian group with basis {${a_\alpha}$} then given the elements {${y_\alpha}$} of an abelian group $H$, there are homomorphisms $h_\alpha : G_\alpha \to H$ such that $h(a_\alpha)=y_\...
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Free object is a free group in the category of groups
I have a question and would appreciate a clear answer.
Firstly, I will provide an introduction regarding my understanding, and then I will ask my question.
Let's begin with the definition of a ...
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Lee Mosher book definition of a tree.
I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.
I am wondering how is this true, can someone explain this to me please?...
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