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.
2,963
questions
2
votes
0
answers
68
views
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 ...
3
votes
0
answers
38
views
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 ...
- 5,031
2
votes
1
answer
90
views
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 $G$-...
- 2,416
0
votes
0
answers
25
views
$G$ 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)...
- 51
0
votes
0
answers
64
views
$\mathfrak{sl}(n,\mathbb{C})$ acts transitively on $\mathbb{C}^n$.
I want to show that the Lie algebra of all trace-$0$ complex matrices acts transitively on $\mathbb{C}^n$, so the standard representation is irreducible. Let $v=\sum_i a_ie_i$, then the matrix
$$
A_{...
- 141
1
vote
1
answer
163
views
Group action of direct product group $G \times H$ when groups $G$ and $H$ do not commute
I'm studying direct product groups actions. Usually, for a group $G \times H$ acting on a set $X$ one takes the group action $\cdot_{G \times H}$ to be the natural $(g \times h)(x) = g \cdot_G (h \...
- 55
1
vote
1
answer
103
views
Is the quotient space of a homogeneous space induced by a free group action s.t. the quotient map has a right inverse homogeneous?
Given is a topological space $X$ and a group $G \leq$ Aut($X$) with the property: for $\lambda \in G$ and $x \in X$, $\lambda(x)=x \Rightarrow \lambda=id_X$, also satisfying that the quotient map $q:X ...
- 73
1
vote
0
answers
8
views
For $\varphi:< S > \times X\to X$ and home. $h:X\to X$, what is size $d(\varphi(s,x), \psi(s, x))$? $\psi$ generated by $h\circ \varphi_s$?
Let $\varphi:G\times X\to X$ be a continuous action of finitely generated group $G=\langle S \rangle$ on compact metric space $X$. Also, assume that $h:X\to X$ be a homeomorphism with $d(h(x), x)<\...
- 1,233
0
votes
0
answers
29
views
Every continuous action $\varphi$ on $S^1\cup \{(x, 0): x\in[-1, 1]\}$ does have $(-1, 0)$ as fixed point
If $X=S^1\cup \{(x, 0): x\in[-1, 1]\}$, then for every homeomorphism $F:X\to X$, $F((-1, 0))=(-1, 0)$. This implies that if $\varphi:G\times X\to X$ is a continuous action, then $\varphi(g, (-1, 0))= (...
- 1,233
1
vote
0
answers
34
views
Transport of group actions via homeomorphisms
Let $G$ be a topological group acting on the topological space $X$ and let $\phi\colon X\xrightarrow{\simeq} Y$ be a homeomorphism between the spaces $X$ and $Y$. Can I induce a $G$-action on $Y$ by $\...
0
votes
0
answers
18
views
Primitive maximal subgroups of $S_{n}$
I notice the following result in GTM163.
Consider $S_{n}$ act on $\lbrace 1,2,\cdots,n \rbrace$ in a natural way. Then the maximal subgroups $M$ of $S_{n}$ fall into three classes:
$(i)$ (intransitive)...
- 11
2
votes
0
answers
29
views
A question about coprime action on a 2-group
I'm stuck somewhere in the following problem in Isaacs Finite Group Theory [4D.4], I would appreciate if you could help:
Problem: Let $A$ act via automorphisms on $G$ , where $G$ is a $2$-group and $A$...
- 407
0
votes
0
answers
41
views
Generalisation of the notion of normalizer
Context
Let $G$ be a Lie group, $K$ a proper subgroup of $G$ and $H$ a proper subgroup of $K$ i.e. $H\subset K\subset G$.
The Lie group $G$ acts on the left on a vector space $V$. Let $x$ be a ...
4
votes
0
answers
52
views
A Question About Coprime Actions
I'm dealing with the following problem in Isaacs Finite Group Theory [4D.3], I would appreciate if you could help:
Problem: Let $A$ act via automorphisms on $G$ , where $(\vert G \vert, \vert A \vert )...
- 407
1
vote
0
answers
36
views
Quotient of projective group scheme by a finite group action
Let $X$ be a projective group scheme, over some base $S$, and let $G$ be a finite group acting on $X$ by $S$-isomorphisms.
I would like to understand if/when the quotient $X/G$ is representable by a ...
- 1,178
0
votes
1
answer
69
views
What does it mean that homogeneous spaces "look the same everywhere"?
In many places, including Wikipedia, a homogeneous space is informally described as "a space that looks the same everywhere, as you move through it, with movement given by the action of a group&...
- 4,311
1
vote
1
answer
39
views
$|\textbf{N}_G(H):H|$ is the number of right cosets of $H$ in $G$ invariant under right multiplication (by $H$)
Here's a problem form I. Martin Isaacs Finite Group Theory (exercise 1A.10.(a)):
Let $H$ be a subgroup of $G$. Show that $|\textbf{N}_G(H):H|$ is the number of right cosets of $H$ in $G$ invariant ...
- 523
1
vote
1
answer
36
views
If $X$ is a finite $G$-set and $|g|=p^n$ with $n\ge 1$, show that $|X|=|X^{g^{p^{n-1}}}|\pmod{p^n}$.
Let $G$ be a group and let $X$ be a finite $G$-set.
Assume that $g\in G$ is of finite order $p^n$, where $p$ is a prime number and $n\ge 1$.
Let $h=g^{p^{n-1}}$.
Show that $|X|=|X^h|\pmod {p^n}$,
...
- 2,387
1
vote
1
answer
36
views
Is there are name for this class of multilinear functions related to the symmetric and alternating classes?
Two famous classes of multilinear functions are the symmetric and alternating multilinear functions, which satisfy for each $\sigma \in S_n$, $\sigma f = f$ and $\sigma f = \text{sgn}\sigma f$, ...
- 2,169
1
vote
1
answer
68
views
Why are quotients by free group actions "well behaved"
Let $M$ be a smooth manifold and $G$ a Lie group, then if $G$ acts smoothly, freely, and properly on $M$ it is a well known result that the quotient $M/G$ is a smooth manifold. In the context of ...
- 2,073
3
votes
1
answer
37
views
Number of copies of irreducible representations for transitive actions
Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then $|X| \le |G|$ since $|X| \big| |G|$ by the orbit-stabilizer theorem. Let $\pi_i;\ \ 1\le i \le m$ be irreducible ...
- 1,953
3
votes
0
answers
44
views
Subsets Whose Translates Under Group Action Are $\subseteq$-Ordered
Suppose $X$ is a space, and $G$ a topological group acting continuously upon it. I'm interested in those closed sets $A \subseteq X$ whose translates under each $g \in G$, up to equivalence, are ...
- 3,252
0
votes
0
answers
49
views
GL(2,3) acting on $X$={{$\pm A$}, {$\pm B$}, {$\pm C$}}$
I am trying to solve this problem.
Let $G$ be $GL_2(\mathbb{F}_3)$ and $$A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, B = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, C=\begin{...
- 47
0
votes
0
answers
33
views
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 ...
- 363
28
votes
3
answers
1k
views
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 ...
- 283
1
vote
0
answers
35
views
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 $\varGamma$ ...
- 41
4
votes
1
answer
121
views
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 ...
- 163
-2
votes
1
answer
58
views
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 ...
- 37
3
votes
3
answers
148
views
Why does $S_n$ act on $\mathbb{R}[x_1,\dots,x_n]$ but not on $\mathbb{R}^n$?
I'm working on a pair of unassessed course problems,
Show that $S_n$ acts on polynomials in $n$ variables by permuting variables.
For $\sigma\in S_n$ and $v\in\mathbb{R}^n$ define $$\sigma\star(c_1,\...
- 1,999
2
votes
0
answers
40
views
Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.
Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.
Let $$D_{2n}=\langle a,b:a^n=b^2=1,bab^{-1}=a^{-1}\rangle $$ be the dihedral group of order $2n$ and $...
- 117
1
vote
1
answer
41
views
Discrepancy in the definition of a module over a group
Let $G$ be a group. A G-module M is defined as an abelian group on which $G$ acts through the map
$ G \times M \to M$
where
$ (g, m) \mapsto g \cdot m$
This action satisfies
$g \cdot (m + m') = g \...
- 4,911
3
votes
1
answer
44
views
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 ...
- 117
0
votes
1
answer
24
views
Group Action Transitive on Orthonormal Bases of Poincare Upper Half-Plane
I'm working on Problem 3-7 of Lee's Introduction to Riemannian Manifolds:
Let $\mathbb{U}^2$ denote the upper half-plane model of the hyperbolic plane (of radius 1), with the metric $\breve{g} = (dx^...
- 387
2
votes
1
answer
132
views
Understanding the Quotient Space of $\mathbb{R}^{n\times m}$ by the Orthogonal Group $O(n)$
I am interested in exploring the structure of the quotient space formed by the action of the orthogonal group $O(n)$ on the space of $n\times m$ real matrices $\mathbb{R}^{n\times m}$ and $m>n$. ...
- 321
0
votes
1
answer
51
views
Is it possible that a group action just acts on the first component of Cartesian product of two groups? [closed]
Let $X$ and $Y$ be different groups. Let $G$ be a group acting on $X$. Is that possible that $G$ act on $X \times Y$ by $g \cdot (x,y) = (g \cdot x,y)$?
I know it sounds silly and unnatural, but for ...
- 149
0
votes
0
answers
86
views
Conjugation as a group action
Disclaimer: my question is different from this question.
I am following these math notes on abstract algebra that I found online to teach myself. On page 21 the group action of conjugation is defined ...
- 54
1
vote
1
answer
44
views
Permutations Quotiented by Group Actions on Finite Sets
I want to preface this by saying that I'm learning about combinatorics for the first time and so my terminology may be problematic.
I am trying to figure out how to count the permutations of a set ...
- 349
0
votes
1
answer
38
views
Inner product invariant under group action - meaning?
My professor wrote a proposition where he defined an inner product on a $G$-module $V$. One of the premises was that the inner product was 'invariant under the action of the group', however he did not ...
- 833
1
vote
1
answer
66
views
Functors inducing group actions.
I'm reading Riehl and I have a question about one of her examples. The setup is as follows.
Suppose that $G$ is a group, which we can consider as a one object category $BG$ with formal object $X$. If $...
- 2,377
0
votes
0
answers
37
views
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 ...
- 124
3
votes
1
answer
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, ...
- 68
1
vote
1
answer
104
views
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 ...
- 499
0
votes
0
answers
19
views
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 ...
- 159
1
vote
0
answers
31
views
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 ...
- 2,207
1
vote
0
answers
36
views
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 ...
- 2,059
1
vote
1
answer
47
views
$\tau\in N_{S_A}(H)$ 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 ...
- 1,069
0
votes
0
answers
41
views
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 $G=\mathbb{Z}_2$ ...
- 1,604
3
votes
0
answers
49
views
Transitivity condition for monoid action
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 $M$...
- 6,691
1
vote
0
answers
23
views
Getting generating set from fundumental domain
I'm reading "Groups, Graphs and Trees" by Meier.
In pages 31-34 (which could be found here) He defined fundumental domain for group action on graphs as
Defintion Let $G\curvearrowright \...
- 46
2
votes
1
answer
91
views
Is the universal cover of a classifying space always $\mathbb{R}^n$ if it is finite dimensional?
If $G$ is a discrete group, and if $BG$ can be represented by a (finite dimensional) manifold, is it always true that $EG$ can be chosen to be $\mathbb{R}^n$, for some $n?$
I'm guessing the answer is ...
- 6,211