Articles of group actions

If a subgroup has smallest prime index, then it is normal

This question already has an answer here: Normal subgroup of prime index 6 answers

Does every homogeneous metric space have a notion of translation?

A homogeneous metric space $(X,d)$ is one where the isometry group $Iso(X)$ acts transitively on $X$. (See this answer to a previous question of mine; note that this is a special case of the notion of homogeneous space.) One consequence of being a homogeneous metric space is that the “local isometry groups”, denoted $Iso(X,x)$ (the […]

Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. So according to the article $T$ should be $(U,V)\cdot X$ $UXV^T\in \mathbf{R}^{n\times n}$; therefore, $\mathbf{R}^{n\times n}$ is invariant under the group action. Why does the […]

Let $G$ act such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Then the Sylow $2$-subgroups acts regular on certain orbits

Let $G$ be a finite permutation group on $\Omega$ acting nonregular and transitive such that each nontrivial element fixes at most two points of $\Omega$. Suppose that for $\alpha \in \Omega$ the stabilizer $G_{\alpha}$ has even order and $|\Omega|$ is even too. Let $S \in \mbox{Syl}_2(G)$ and suppose $S_{\alpha} \ne 1$, but $S \nleq G_{\alpha}$ […]

In what sense is this action of $\mathbb R$ on $T$ lifted to an action of $\pi_1(T)\times\mathbb R$ on $\mathbb R^2$?

I am reading the paper “Calculating the fundamental group of an orbit space” by M A Armstrong where he states the following – Let $\mathbb R$ act on the torus $T\cong S^1\times S^1$ by $$r\cdot(e^{2\pi ix},e^{2\pi iy})=(e^{2\pi i(x+r)},e^{2\pi i(y+r\sqrt{2})})$$ This action lifts to an action of $G=\pi_1(T)\times\mathbb R$ on $\mathbb R^2$ which has the same […]

The action of $PSL_2(\mathbb{R})$ on $\mathbb{H}$ is proper

Consider the upper half-plane $\mathbb{H}:=\{z\in \mathbb{C}:\Im (z)>0\}$ with the hyperbolic metric, and the group $PSL_2(\mathbb{R})=SL_2(\mathbb{R})/\{\pm I_2\}$, where $SL_2(\mathbb{R})$ is the set of $2\times 2$ real matrices with determinant equal to $1$, which acts on $\mathbb{H}$ by Möbius transformations. I’d like to show that this action is proper, i.e. For any compact set $P\subset \mathbb{H}$ there […]

Is there an example of a non compact, semisimple, amenable Lie group?

By semisimple I mean the real Lie algebra of $G$ is semisimple. I guess there is not but I can’t formulate a rigorous argument.

The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$

Let $G$ be a permutation group acting transitively on $\Omega$ and suppose $N \unlhd G$ is a normal subgroup of $G$. Assume that for $g \in N_g(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\alpha}^g = 1 $$ for some point stabilizer $G_{\alpha}$. Also assume that every nontrivial element either has no fixed point or exactly $p$ […]

Group Actions: Orbit Space

Given a group action $G\curvearrowright X$. Consider the orbit space: $\pi:X\to X/G$ Do continuous group actions correspond to open projections, i.e.: $$l_g\in\mathcal{C}(X)\quad(g\in G)\iff\pi(U)\in\mathcal{T}_{X/G}\quad(U\in\mathcal{T}_X)$$ (Note that this is a slightly different more appropriate version of continuous group actions.) Certainly, continuous group actions give rise to open projections since: $$\pi^{-1}(\pi(U))=\bigcup_{u\in U}Gu=GU=\bigcup_{g\in G}gU=\bigcup_{g\in G}l_g(U)\in\mathcal{T}$$ Surely, the converse may […]

Transitive action of normal subgroup of the alternating group

everyone! Would anyone be willing to give me any sort of help with the following question? Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the action of $N$ on $\{1,2,…,n\}$ is transitive. Let me stress that one is NOT allowed to use the fact that […]