# Questions tagged [group-actions]

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 $2n$ act 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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### Why is it worth studying groupoids if they are so similar to groups?

I know that in the pure algebraic context, connected groupoids are equivalent to groups. Moreover, connected groupoids can be seen as action groupoids. Consequently, any abstract groupoid is ...
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### Other things with operators

There is the notion of "groups with operators" which signifies a group $G$ together with a morphism of sets $X \to \operatorname{End}(G)$. It is easily observed that the endomorphism monoid ...
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### Proof of $\operatorname{Aut}_G(G/H)\cong N_G(H)/H$

Here $G$ is a group with subgroup $H$, and we let $G$ act on $G/H$ by left multiplication. Correspondingly, $G/H$ is a left $G$-set and the set $\operatorname{Aut}_G(G/H)$ denotes the set of all $G$-...
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### $G$ transitive, some Stab$_G(x)$ transitive $\implies$ all Stab$_G(x)$ transitive [duplicate]

I'm currently working through Dobson, Malnič, Marušič's Symmetry in Graphs and am stuck on the following problem (from Lemma 4.4.3): Let $G$ be a (finite) group which acts transitively on a (finite)...
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### Cartesian product of transitive group actions, stabilizer groups

Consider a finite group $G$ and subgroups $\{H_i\}_{i=0,\ldots,n-1}$ (defined up to conjugacy) such that $G$ acts transitively on each $G/H_i$ [edit: in the canonical way]. Consider the induced action ...
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### Orbit stabiliser theorem as an analogue to first isomorphism theorem

The notes I'm using to study group theory make a remark that another appropriate name for the "orbit stabiliser theorem" is the "first isomorphism theorem for group actions". For ...
1 vote
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### Is the orbit space of discrete group act continuously and properly on manifold with boundary is still a manifold with boundary?

This question was inspired by Lee's Introduction to Topological Manifold problem 12-22 in chapter 12. Here is the original problem: Give an Example of a manifold $M$ and a discrete group $\varGamma$ ...
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### Toric vector bundle - Klyachko's classification

I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra). I am ...
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### Continuous group action and convergent sequences. [closed]

I would like to see if the following statement is true in general. Statement: Let $G$ be a topological group acting on a topological space $X$ continuously. Suppose that a sequence $g_n \in G$ and an ...
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### Functor description for topological group action

Since we know that a group action $F: G\times S\rightarrow S$ can be described by a functor $F : G\to \mathrm {Set}$, See here. I would like to know how can we describe a topological group action with ...
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### Redfield-Polya enumeration, but the colors don't matter

Is there a version of Redfield-Polya enumeration with the added condition that you don't care which color is which? An illustrative example is: Count edge-colorings of $K_4$ modulo the group action of ...
118 views

### How do group actions connect to group presentation?

I am editing this question so that it is clearer for other users with a similar question, although I consider the answers I received sufficiently explanatory for my purposes. I am an undergraduate, ...
1 vote
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### Properties of the obvious action of $Aut(G)$ on $G$

Question: Let $G$ be a finite group. Let $Aut(G)$ be the group of automorphisims of $G$. Consider the group action $\phi:Aut(G)\times G\to G$ where $\phi(\sigma,g)=\sigma(g)$. Assume $G$ has exactly ...
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### Homogenous geodesic metric spaces

Consider a metric space with a path between any two points, so a real line segment of some length between them, and the length of this line is the same as the distance between the two points in the ...
1 vote
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### In the regular wreath product $W = H \wr G$, every subgroup of $G$ is the centralizer of some element

This is Exercise 3A.9 - (b) from M. Isaacs' "Finite Group Theory". It goes as follows: Let $W = H \wr G$ be the regular wreath product and let $C \leq G$ be an arbitrary subgroup. Show that ...
1 vote
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### A Lie group $G$ be acting on a Riemannian manifold $M$. Is the map $G\times M\to \mathbb{R}$ given by $(g,x)\mapsto \lVert dg_x\rVert$ continuous?

The norm involved is the operator norm $\lVert T\rVert=\sup\{\lvert T(x)\rVert:|x|\leq 1\}$. Since $g$ is smooth, of course $x\mapsto \lVert dg_x\rVert$ is continuous. But I am having little trouble ...
1 vote
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### $\tau\in N_{S_A}(H)$ then $\tau$ stabilizes both $F(H)$ and $A-F(H)$

(I have seen this question:Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$, but the notation in both the question and the comment was unfamiliar to ...
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### When are homeomorphic $G$-spaces isomorphic?

Let $X,Y$ be $G$-spaces ($G$ a topological group). In general $X$ and $Y$ can be homeomorphic as topological spaces without being isomorphic as $G$-spaces. For example $X=Y=S^1$ and $G=\mathbb{Z}_2$ ...
I am interested in a simple condition for a continuous monoid on a topological space to be topologically transitive. My setting is as follows: Let $X$ be a $2$nd countable topological space and let $M$...