All Questions
Tagged with free-groups normal-subgroups
29
questions
1
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56
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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 ...
- 59
0
votes
1
answer
95
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When is the Schur multiplier complemented in $R / [F, R]$?
A normal subgroup $K \trianglelefteq G$ is complemented if there is a subgroup $H \le G$ such that $H \cap K = \{e\}$ and $G = HK$, i.e., $G$ is the semidirect product of $H$ and $K$ (Wikipedia).
Let $...
- 1,954
5
votes
1
answer
415
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Normal subgroup generated by one element
Let $F$ be a free group of rank at least two and $\alpha,\beta$ be two elements in $F$. Let $N(\alpha)$ (resp. $N(\beta)$) be the normal subgroup generated by $\alpha$ (resp. $\beta$); i.e., $N(\alpha)...
- 145
-2
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47
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If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$ [closed]
This is probably a basic fact of group theory but I am not able to prove it:
Let $F_2$ be the free group generated by 2 elements and $H,K$ be two normal subgroups of $F_2$. If $F_2/H$ is isomorphic ...
- 861
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1
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78
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Question about f.g. normal subgroups of f.g. free groups
Suppose $H$ is a finitely generated normal subgroup of a finitely generated free group $F(X)$ with basis $X = \{ x_1 , \dots , x_n \}$. Fix $1 \leq i \leq n$.
Is it true that the set of right cosets $\...
- 303
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votes
1
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149
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Describe all path-connected $3$ fold covers of $X=S^1 \vee \mathbb{R}P^2$
Describe all path-connected $3$ fold covers of $X=S^1 \vee \Bbb{R}P^2$ (please justify why your list is exhaustive). Which are regular (i.e. normal) and why?
I get that $X$ has a universal cover, ...
- 4,510
2
votes
1
answer
76
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Problem handling free groups in algebraic topology
Trying to compute the fundamental group of a topological space $X$ I have come to the equality
$$\pi_{1}(X)\cong\frac{\ast_{i=1}^{n}\mathbb{Z}}{G}$$
where $\ast$ means taking the free product ($n$ ...
- 188
3
votes
1
answer
206
views
Normal closure in free group
Let $G=\pi_1(\mathbb{C}-\{z_1,z_2\})$ be the fundamental group of the plane punctured twice. There exist two homotopy classes of loops $\mathfrak{g}_1=[\gamma_1]$ and $\mathfrak{g}_2=[\gamma_2]$ such ...
- 186
2
votes
1
answer
77
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Let $F$ be freely generated by $\{a, b, c\}$. Prove that $F/N$ is free with basis $aN, bN$, where $N$ is the normal subgroup of $F$ generated by $c$
My question stems from the following exercise in free groups:
Let $F$ be freely generated by $\{a, b, c\}$ and let $N$ be the normal subgroup of $F$ generated by $c$. Prove that $F/N$ is freely ...
- 2,663
2
votes
1
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255
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For $N\unlhd G$ for $\mathfrak{B}$-group $G$ with $G/N$ a free $\mathfrak{B}$-group, show $\exists H\le G$ with $G=HN$ and $H\cap N=1$
This is Exercise 2.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". In the book, maps are evaluated from left to right. According to Approach0, the exercise is new to ...
- 45.7k
3
votes
0
answers
36
views
Finitely presenting implies finite presenting subset?
I recently had the following problem as a homework assignment.
Let $S$ be a finite generating set of a group $G$. Suppose that G is finitely presenting over $S$, and let $R$ be any set of relations ...
- 6,610
2
votes
1
answer
342
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Quotient of amalgamated free product by normal subgroup
I was given the following question in my homework:
Given $Γ =G_1∗_HG_2$, suppose that $H$ is a normal subgroup in both
$G_1$ and $G_2$.
(a) Show that $H$ is then a normal subgroup in $Γ$.
(b) Show ...
- 1,399
0
votes
1
answer
49
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Relative weight of normal subgroups of free groups?
I'm not sure what the standard terminology for the following setup is, so I will make up some (but I'd love any references to what the correct terminology is). Let $N$ be a normal subgroup of a group ...
- 5,311
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1
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206
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Quotient of a free group by a subgroup of the commutator [closed]
Let $F$ be the free group on finitely many generators $x_1,\dotsc,x_n$. Let $[F,F]\subseteq F$ be its commutator, so we know $F/[F,F]\cong \mathbb{Z}^n$, and let $N\subseteq F$ be a normal subgroup ...
- 759
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1
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203
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When is a quotient group of a free group finite?
Let $I=\{\alpha\}$ and $k\in\mathbb{N}$. Consider the free group $F(I)$ constructed on $\{\alpha\}$. Let $\phi_\alpha$ be the canonical homomorphism of $\mathbb{Z}$ into $F(I)$. Let $r=\phi_\alpha(1)^...
