# Tagged Questions

Use with the (group-theory) tag. Groups describe the symmetries of an object through their actions on the object. For example, Dihedral groups of order $2n$ acts on regular $n$-gons, $S_n$ acts on the numbers $\{1, 2, \ldots, n\}$ and the Rubik's cube group acts on Rubik's cube.

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### Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
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### Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
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### Dihedral Group question involving $D_{4n}$

Let $D_{4n}$ be the dihedral group of the order $4n$. Prove that $D_{4n}/\langle T^n \rangle$ is isomorphic to $D_{2n}$. We tried to configure an action on the diagonals of the $n$-gon and prove ...
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### actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
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### Are there different combinatorial species with the same symmetry type?

First off: for my purposes, let $\sf B$ be the category of finite sets with bijections, and ${\sf B}_n$ the subcategory of sets with cardinality $n$, and define a combinatorial species to be a functor ...
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### Calculate the homology group of $S^3/G$, an Harvard qualifying exam problem with “unclear” solution

Problem Suppose that $G$ is a finite group whose abelianization is trivial. Suppose also that $G$ acts freely on $S^3$. Compute the homology groups (with integer coeffcients) of the orbit space ...
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### Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
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### What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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### Is the orbit map for a group action closed in this case?

Suppose a compact Lie group $G$ acts on a manifold $M$ and let $\pi : M \rightarrow M/G$ be the orbit map. Can I say that $\pi$ is closed map? If $C \subseteq M$ is a closed set in $M$ then I only ...
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### G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
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### Group Actions: Verify a Bijective Correspondence

This is an old exam problem: Given an action of $G$ on $X$, we can define $\varphi: G \to S_X$ by the rule $\varphi(g) = \sigma_g$, where $\sigma_g$ is left multiplication by $g \in G$. Prove that ...
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### Does transitive imply it's the entire symmetric group

Let $G$ denote a finite group and recall that $G$ acts transitively (on itself) if and only if for all $x,y \in G$ there is a $g \in G$ such that $gx = y$. I am wondering if transitive may imply that ...
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### The Stabilizer of the coset for the action of G on $G/H$ by left multiplication.

Let $H$ be a subgroup of $G$. What is the stabilizer of the coset $aH$ for the action of $G$ on $X=G/H$ by left multiplication? So, I think I've done this one correctly: The Stabilizer is of the ...
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### action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
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### Continuous action on tensor product

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple ...
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### Group orbit computations

Can someone please verify these? Compute the orbit of a vertex of the square under the tautological action of $D_4$. Compute the orbit of a point $x \in G$ under the action of $G$ on itself by ...
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### the largest prime number $p$ dividing $\left|G\right|$ is also divide $\left|X\right|$.

Let $G$ be a finite simple group which acts on finite set $X$ non-trivially. The goal is that the largest prime number $p$ dividing $\left|G\right|$ is also divide $\left|X\right|$. I know ...
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### Transfer homomorphism in transformation groups

I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups. But suppose ...
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### In which condition the restriction of the orbit map is open?

Let $(X,G)$ be a group action, must the restriction of the orbit map be an open mapping? In other words, $\phi: G \times X \to X$ is a group action, for any $x\in X$, $\phi_x:G \to Gx$ is an open ...
### $p$-group acting on a finite set
Let $G$ be a $p$-group. Prove that if $G$ acts on a finite set $X$ and $p$ does not divide $|X|$, then $X$ contains some element that is fixed by every element in $G$. Any thoughts? I'm stumped ...