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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Is there a general way to find the inverse of an automorphism of the free group? [closed]
If we describe an automorphism of the free group (on n generators) by where it sends the generators, is there some kind of algorithm to find the inverse automorphism? I am particularly interested in ...
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Relative divisibility of derived subgroup of free group [closed]
Let $F$ be a free group (possibly on an infinite set) and let $[F,F]$ denote its derived subgroup.
Can there be a $w \in F \setminus [F,F]$ and an $n>0$ such that $w^n \in [F,F]$?
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Transition matrix associated to representative of element in $Out(F_n)$
There is a notion of transition matrix associated to elements in $Out(F_n)$ from Bestvina and Handel's paper that I am a little bit confused.
Let $\Phi\in Out(F_n)$ and $\phi:\Gamma\to\Gamma$ a ...
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Questions on the $\hom$ Functor and Free Groups
This question arises while learning about the $\hom$ functor. My algebra background is not that strong, so here is my question:
Let $G$ be a free group, and let $f\colon G \to G$ be a group
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Relative singular chains basis
If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
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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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$SO_3(\mathbb Q)$ 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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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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Free group on a set $X$ is generated by $X$, why?
The definition of free group given in class was: Given a non-empty set $X$ and an inclusion $i:X\rightarrow F$. We say that $F$ is a free group on $X$ if, given any group $G$ and a set map $\varphi:X\...
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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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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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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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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 ...
CommunityBot
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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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