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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Faithful actions of the free group on two generators
I'm trying to come up with faithful actions of the free group on two generators, $G$.
By Cayley's theorem, $G$ acts faithfully on itself by left (or right) multiplication. There are also variants of ...
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Alternative proof that adding an additional square root of the identity to the free group on one generator results in a non-Abelian group
I'll be using additive notation for groups regardless of whether the group in question is Abelian or not.
I am thinking about the consequences of adding propertyless square roots of existing elements ...
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Adding commutation rules to a free group?
I'm interested in knowing how, given a set $S$, one can modify the free group $F(S)$ by adding one or more commutation rules and get a new group. For instance, adding the commutation rule:
$$\forall x,...
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Checking the image of mapping class in $\text{Aut}(F_{2g})$ stabilizes boundary curve
Overview: the mapping class group maps into $\text{Aut}(F_{2g})$ and its image stabilizes the surface relation. I am trying to check this for a specific example and am doing something wrong.
The ...
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A free generating set $S$ of a free group $F$ has the smallest cardinality of the generating sets.
I'm reading Clara Löh's book "Geometric group theory, an introduction" and i'm going through the free groups section. She stated the following:
Let $F$ be a free group.
Let $S\subset F$ be ...
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Can the following proof calculus show that any finitely presented free group is free?
If a finitely presented group is free, will it always have a proof in the proof calculus outlined in this question that it is free?
I recently saw this question.
I tried to show that the group was ...
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Pushout of Z isomorphic to Z
$\mathbb{Z}\xleftarrow{\text{id}}\mathbb{Z}\xrightarrow{\text{x2}}\mathbb{Z}$
How do we prove this group pushout is isomorphic to $\mathbb{Z}$?-
I know we can take $\langle x|\rangle$ as a ...
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Presentation of Product Group
Here is the question I have been working on:
If $G_1 = \langle X_1 : R_1\rangle$ and $G_2 = \langle X_2 : R_2\rangle$, supply a presentation for $G_1 \times G_2$.
Deduce that, if $G_1$ and $G_2$ are ...
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Ping Pong Lemma
In the book "Groups, Graphs and Trees" by John Meier, Lemma 3.10 is the Ping Pong Lemma. He uses different assumptions than for example Wikipedia. Namely, he states that
Let $G$ be a group ...
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How do we generate the loop $ba$ from the loops $a^2,b^2$ and $ab\ $?
In the second diagram a $2$-sheeted connected covering of the figure eight has been described. The image of the fundamental group of the covering space has the generators $a^2, b^2$ and $ab$ as ...
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Has anyone studied differential equations on $\mathbb{Z}F_n$ defined using Fox derivatives?
I am looking for a reference, if it exists, to the study of differential equations defined using Fox derivatives over the group ring, say, of a free group. Is this a topic which has been studied ...
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Is group defined by action on $\mathbb{R}$ a free group?
Assume that I have two bijective functions $f,h:\mathbb{R}\to\mathbb{R}$ such that $f(h(x))\neq h(f(x))$ as well as $f(x)\neq x \neq h(x)$ for all $x\in\mathbb{R}$. Additionally, I consider the group $...
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Show a free group has no relations directly from the universal property
The free group is often defined by its universal property. A group $F$ is said to be free on a subset $S$ with inclusion map $\iota : S \rightarrow F$ if for every group $G$ and set map $\phi:S \...
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The fundamental group of closed orientable surface of genus 2 contains a free group on two generators
Let $S$ be the closed orientable surface of genus $2$. It is well known that its fundamental group is given by $$ \pi_1(S)=\langle a,b,c,d:[a,b][c,d]=1\rangle.$$
How can we show that this group has a ...
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Proving that 2 formulations of the Universal Property of Free Groups are equivalent
The present question is inspired by an answer to this question of mine on applications of Category Theory in Abstract Algebra.
One of the answers stated that the universal property of free groups ...
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