3
votes
0answers
39 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
2
votes
1answer
89 views

All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
0
votes
1answer
51 views

Plane rotation homomorphism

I want to show that the following set (given group structure with matrix multiplication) $$D=\left\{ D(\theta)=\begin{pmatrix} e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2} \end{pmatrix}, 0 \leq ...
14
votes
1answer
366 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
3
votes
1answer
140 views

Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?

I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following: If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of ...