All Questions
Tagged with free-groups definition
15
questions
2
votes
2
answers
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Lee's definition of free abelian group
In Topological manifolds (Book), Lee defined the free product then free group by construction. i.e. defining a word and making set of all words a group. After that he defined Free Abelian Group in ...
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Motivation for definition of free group?
Let $S$ be a set and $F_S$ be the equivalence classes of all words that can be built from members of $S$. Then $F_S$ is called the free group over $S$.
I don't understand the motivation for this ...
- 37.2k
1
vote
1
answer
339
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Royal Road to Free Groups and Free Products
This question is more about strategy, which can be used when developing group theory, then about a particular proofs.
$
\newcommand{GRP}{\mathsf{GRP}}
\newcommand{SET}{\mathsf{SET}}
$
One way to ...
- 1,879
1
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Free monoid of natural numbers excluding zero
Let $U$ be a set of finite sequences like $\{1,1\cdot1,1\cdot2,\dots,1\cdot 2\cdot3,\dots\}$, i.e. there is no $0$ element in any sequence and all sequences start from $1$.
Can this set be defined as ...
- 139
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5
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What is a simple example of a free group?
Can someone give me a simple example of a free group with a basis, given the definition below? I don't think I'm understanding the definition clearly.
For example if $F= (\Bbb Z, +)$, $X = \{0\}$, $\...
- 4,912
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1
answer
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Definition of infinite cyclic subgroup?
Let $B$ be a subset of an (additive) abelian group $F$. Then $F$ is free
abelian with basis $B$ iff the cyclic subgroup $\langle b \rangle$ is infinite cyclic for each $b \in B$ and $F = \sum_{b \in B}...
- 4,912
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3
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Trying to understand the definition of a free Abelian group.
Let $B$ be a subset of an (additive) abelian group $F$. Then $F$ is free abelian with basis $B$ if the cyclic subgroup $\langle b \rangle$ is infinite cyclic for each $b \in B$ and $F=\sum_{b \in B} \...
- 4,912
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1
answer
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Formal construction of free groups and objections in arguments
For simplicity, consider $X=\{a,b\}$. Let $Y$ be another set in bijection with $X$, and write its elements to be $a^{-1},b^{-1}$.
Let $W(X)$ be the collection of all words in $a,b,a^{-1},b^{-1}$, ...
- 10.3k
3
votes
3
answers
547
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Strange question about free abelian group
I was given the following question for homework, but it makes no sense to me.
Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set $...
user153312
10
votes
3
answers
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Precise definition of free group
I have seen the definition of a free group go like this:
Let $S = \{s_i : i\in \mathbb{N} \}$ be a countable set. Let $S^{-1}$ be the set $\{s_i^{-1}: i\in \mathbb{N}\}$. Here one is to understand $...
- 3,309
2
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1
answer
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What is 'free algebra'?
I've been googling the definition of it, and it seems like somehow it's related to a polynomial ring.
But I still quite don't get it.
Is a free algebra just a free group with additional operation (...
- 41
1
vote
1
answer
295
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What is a "right" automorphism?
Let $B_n$ be the braid group with $n$ strands and let $F_n$ be the free group of rank $n$ generated by $x_1,\ldots,x_n$. The classical Artin Representation Theorem reads:
If an automorphism of $F_n$ ...
- 5,451
1
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1
answer
430
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"Length" of an element in a free group
Is there any universally agreed definition of "length" (or "width", or whatever term) of an element in a free group $F_n(x_1,\cdots,x_n)$? Intuitively, I would like the length of $1$ to be $0$; the ...
- 5,451
11
votes
4
answers
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Is the free group on an empty set defined?
I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.
- 111
9
votes
1
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Free group and universal property
I'm trying to understand universal properties. An example is the definition of a free group (as I understand it so far):
Revised definition:
A free group $F_S$ over a set $S$ is a pair $(g,F_S)$ ...
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