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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Every graph morphism that is an immersion and surjective on the fundamental group is a homeomorphism
Here, we consider graphs as 1-dimensional CW complex and a graph morphism is a map sending vertices to vertices and $[f(a),f(b)]=f([a,b])$ where $[a,b]$ represents an edge connecting vertices $a,b$. A ...
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Direct sum of free abelian group and quotient of abelian group by subgroup
I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem:
Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
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$SO_3(\mathbb Q)$ contains a free group using 5-adic numbers
I am trying to show that $SO_3(\mathbb Q)$ contains a free group using 5-adic numbers, and more precisely using the matrices $M_1=\left(\begin{array}{ccc}1 &0&0\\0&\frac 35&-\frac{4}5\\...
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Can we construct a free structure on a non associative algebraic structure.
For any set we can construct a free group on it. Also for non associative structures like Lie algebra, Lie ring we may construct free structures, but these are non associative structures and having ...
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Technique for showing a group is not free?
The specific case I present here is much less important than the general question. I have two matrices: $$
p = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & -\frac{1}{2} \\
0 &...
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"Almost Retractible" Abelianizations of Groups
I have two related questions.
Is there a name for a nonabelian group $G$ whose abelianization is $\bigoplus_{i=1}^n \mathbb{Z}/p_i\mathbb{Z}$ such that for each $i$ there is an element $g$ whose ...
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Tiling of a tree to show that a group acting freely on a tree is free
Let me start giving some context:
Let $G$ be a group acting freely on a tree $T$. Let $T'$ be the barycentric subdivision of $T$ (that is, the graph obtained by placing a new vertex at the center of ...
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If a graph map is an immersion, then the induced homomorphism on fundamental groups is injective
So I was reading some Geometric group theory and came across Stalling's folding of graphs. Now I am trying to use the folding idea to prove that every finitely generated subgroup of a free group is ...
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How to prove the free group $F_n$ cannot be generated by $n$ elements as a monoid?
I cannot figure out how to prove that the submonoid of $F_n$ generated by elements $\alpha_1,\dots , \alpha_n$ will never be the whole group.
It clearly is possible to generate the free group on $a_1, ...
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Free product contains the free product of itself with a free group.
I saw this answer and I'm thinking if with this idea we can show that $G_1 \ast G_2 \ast F$ (where $F$ is a free group of countable rank) embedds in $G_1 \ast G_2$ if $G_1$ or $G_2$ has cardinality at ...
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Sorting integers by looking at their prime factorizations
By the fundamental theorem of arithmetic, we know that any positive integer can be uniquely defined by its prime factors. Now, suppose $S_{\infty}$ is the set of all primes, and let $s_i
\in S$ such ...
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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 $\ker\phi$. ...
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