All Questions
Tagged with free-groups combinatorial-group-theory
96
questions
9
votes
2
answers
134
views
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
3
votes
0
answers
50
views
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 ...
- 149
2
votes
1
answer
78
views
Is there an enumeration of finitely presented groups?
I know that the general word problem is undecidable, but is there an effective enumeration of presentations all finitely presented groups generated by $n$ elements in which each isomorphism class of a ...
- 2,697
2
votes
1
answer
142
views
Ways to show that words with exponent sum zero for each generator are elements of the commutator subgroup
Say I have a free group on the generators $X = \{ x_1, x_2, ... , x_k \}$ with $k \geq 2$. I read (in an article) that if I have a word $w$ written in the generators and their inverses, and the ...
-1
votes
1
answer
65
views
Can you determine the order of a generator in this group presentation? [closed]
Given the following group presentation $<x,y|2x+3y=0, 5x+2y=0>$ of an Abelian group, find the order of element x.
My follow up question: Is there a way to determine the order without finding ...
- 140
2
votes
1
answer
75
views
Schreier basis of kernel of $F(G)\to G$ for $G$ a group
Let $G$ be a group and $F(G)$ the free group on $G$ as a set. There is a natural epimorphism $F(G)\to G$ that maps $[\sigma]\in F$ to $\sigma$, let $K$ be its kernel. Is the set $$X=\{[\sigma][\tau][\...
- 1,434
2
votes
0
answers
55
views
Presentation of Product Group
Here is the question I have been working on:
If $G_1 = \langle X_1 : R_1\rangle$ and $G_2 = \langle X_2 : R_2\rangle$, supply a presentation for $G_1 \times G_2$.
Deduce that, if $G_1$ and $G_2$ are ...
14
votes
2
answers
626
views
Show a free group has no relations directly from the universal property
The free group is often defined by its universal property. A group $F$ is said to be free on a subset $S$ with inclusion map $\iota : S \rightarrow F$ if for every group $G$ and set map $\phi:S \...
- 623
1
vote
0
answers
143
views
If $a^{p_1} b^{q_1}\ldots a^{p_n} b^{q_n} = e$ then $S = \{a,b\}$ is not a free generating set of $G = \langle S \rangle$
Let $S = \{a,b\}$ be a generating set for a group $G$. If a non-trivial word in $a, b \in S$ equals the identity $e$ of $G$, i.e.,
$$a^{p_1} b^{q_1} a^{p_2} b^{q_2} \ldots a^{p_n} b^{q_n} = e$$
for ...
- 14.2k
13
votes
2
answers
204
views
Is $\Bbb Z^3$ a one-relator group?
I understand that:
$\Bbb Z^0 = \langle a \mid a \rangle$
$\Bbb Z^1 = \langle a, b \mid b \rangle$
$\Bbb Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$
but is it possible for $\Bbb Z^3$ to be ...
- 623
1
vote
0
answers
150
views
Normal subgroup of fundamental group of Klein Bottle
Let $K$ the Klein Bottle and $\pi_1(K) = \langle a,b \mid b a b a^{-1} \rangle $ be the fundamental group of the Klein bottle. Observe that $\langle b \rangle $ is a normal subgroup of $\pi_1(K)$, ...
- 592
4
votes
1
answer
150
views
Rank of free groups
In Johnson's 'Topics in the Theory of Group Presentations', one can find this theorem after the definition of free groups using the universal property.
Theorem. Free groups of different ranks are not ...
- 315
1
vote
2
answers
70
views
Let $\Bbb Z*\Bbb Z=\langle a,b\rangle$ and $N=\{waba^{-1}b^{-1}w^{-1}:w\in\Bbb Z*\Bbb Z\}.$ Prove $\langle a,b\rangle/N$ is abelian
Let $\mathbb{Z} * \mathbb{Z} = \langle a,b \rangle$ and $$N = \left\{w a b a^{-1} b^{-1}w^{-1}: w\in \mathbb{Z} * \mathbb{Z} \right\}$$ the smallest normal subgroup that contains $\left\{ a b a^{-1} ...
- 592
2
votes
2
answers
212
views
Computation of Amalgamated Product $\mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6.$
I'm trying to compute a amalgamated product $\mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6$.
Let $\mathbb{Z}_4= \langle a\mid a^4 =1\rangle$ and $\mathbb{Z}_6 = \langle b\mid b^6 =1\rangle $, be a ...
- 592
5
votes
0
answers
128
views
Is there a sense in which $(\mathbb{R},+)$ is "free"?
Given a set $S$, the free group on $S$ consists of finite strings of elements in $S$. They can be visualized as paths on the integer grid $\mathbb{Z}^S$ starting at the origin, with the group ...
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