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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50 views

Finite groups whose non-trivial elements have no fixed points

(I first asked this question on MathOverflow, but was recommended to ask here at Mathstackexchange instead.) I am interested in finite groups $G$ acting on a finite set $X$ with the following ...
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1answer
17 views

Identification of the Lie algebra of an isotropy group with the tangent space - stuck with a statement

I think I am stuck with the following statement that I read on the Encyclopedia of Mathematics website regarding Isotropy representations: "If $G$ is a Lie group acting smoothly and transitively on ...
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0answers
20 views

Finite groups whose non-trivial elements have no fixed points [duplicate]

I am interested in finite groups $G$ acting on a finite set $X$ with the following property: (*) fix(g)=$\emptyset$ for all $g\in G\setminus\{1\}$, where fix(g):=$\{x\in X|gx=x\}$ denotes the set ...
0
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1answer
21 views

The action of the orthogonal group $O_n(\mathbb{R})$ on the Stiefel manifold $V_{k,n}(\mathbb{R}) $ .

I'm trying to prove that $O_n(\mathbb{R}) / O_{(n-k)}(\mathbb{R}) \cong V_{k,n}(\mathbb{R})$ where $ O_n(\mathbb{R}) = \left\lbrace A \in M_n(\mathbb{R}) / A A^t = I_{n}\right\rbrace $ is the ...
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0answers
100 views

Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
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0answers
10 views

Showing the action of $SO(p,q)$ in the punctured cone of isotropic vectors is transitive

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
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1answer
42 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
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0answers
30 views

Deriving that center is subset of centralizer(using group actions only)

So if we consider our group $N_G(A)$ and let the group act on the set A $\in$ P(G) via conjugation and consider the kernel we will get precisely the kernel being the centralizer, and since the kernel ...
4
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1answer
68 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
2
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0answers
37 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
0
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0answers
15 views

Action of $SL_2(p)$ on intgers mod p by Möbius transformations

This is simple and may have been asked before, but I couldn't find it. I have been asked to 'Define the action of $SL_2(p)$ (the group of 2 by 2 matrices of determinant 1 with entries in ...
2
votes
1answer
27 views

Image of homomorphism that acts transitively and contains the stabilizer is surjective.

Let $\phi:G_1\rightarrow G_2$ be a homomorphism of groups. Let $G_2$ act on a set $X$, and let $\phi(G_1)$ act transitively on the same set $X$. Finally, let $p\in X$ be arbitrary, and suppose that ...
1
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2answers
39 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
4
votes
1answer
65 views

Quotient space of $\Bbb C^n$ obtained by action of $S_n$

Consider the action of $S_n$ on $\mathbb{C^n}$ given by: $$\sigma(x_1, x_2, \cdots,x_n) = (x_{\sigma(1)}, x_{\sigma(2)}, \cdots,x_{\sigma(n)}).$$ What is the quotient space of $\mathbb{C^n}$ obtained ...
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0answers
60 views

Is this an action of $S_{n}$ on $\mathbb{R}_{n}$?

I am trying to prove that $S_{n}$ acts on $\mathbb{R}_{n}$ with the map $$* : S_{n} \rightarrow \mathbb{R}_{n}, \quad * \left( \sigma, \left( r_{1}, r_{2}, \dots, r_{n} \right) \right) = \left( ...
0
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1answer
27 views

If $G$ acts non-trivially on a set then it has a proper normal subgroup of finite index

The following is an old exam question from a n introduction to group theory course: Let $G$ be a group (possibly infinite) that acts non-trivially on a finite set $X$ - that is there are $g\in ...
2
votes
1answer
37 views

$p$-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 ...
0
votes
1answer
33 views

The stabiliser of $g.x$ is the subgroup $gGxg^{-1}$

Let $X$ a G-set and $x \in X$. Show that for any $g\in G$, the stabiliser of $g.x$ is the subgroup $gGxg^{-1}$. I found this question in the book A Course in Group Theory of J. Humphreys. I was ...
3
votes
0answers
33 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
2
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3answers
59 views

Examples of Non-Faithful Group Actions

I cannot find anywhere a relatively simple example of a non-faithful group action. I feel I understand the definition relatively well, however I can't come up with any ideas for one in my head (and ...
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1answer
65 views

Consider group G acting on a set X

Consider group G acting on a set X Give examples of: a)The action that is transitive and faithful My Answer: Group G under addition acting on a set of integers Z b)The action that is transitive ...
1
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1answer
32 views

Is the saturation of Borel sets Borel?

Problem. Let $G\times X\rightarrow X$ be a continuous action of a Polish group on a Polish space. Let $A\subseteq X$ be Borel. Is the saturation $[A]_{G}:=G\cdot A$ a Borel set? One approach. The ...
3
votes
1answer
72 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
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0answers
61 views

Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
0
votes
0answers
45 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
4
votes
3answers
58 views

is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
1
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1answer
31 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
4
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2answers
150 views

What is a “natural group action”?

Eg. The symmetric group on S acts on S in a natural way, for all sets S. Thanks in advance!
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1answer
45 views

Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states: Let $G$ act ...
1
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1answer
64 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
0
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0answers
32 views

for which values of k is the action of $S_A$ on k-element subsets faithful

I am solving dummit and foote question and I saw this question in one of the chapters. So in order to be faithful we have to have the given action induce distinct permutation on the set, but the ...
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0answers
12 views

cocompact action + finite stabilizers => proper action?

Assume a disctere countable group G acts on a smooth manifold M by diffeomorfisms and (1) M/G is compact and Hausdorff (2) all stabilizers are finite How to prove that the action is proper?
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1answer
30 views

Orbit-Stabiliser Theorem Applied

"The group S6 acts on the group Z6 via σ([a]) = [σ(a)], for σ ∈ S6 and a∈{1,...,6}. A permutation that is also an isomorphism is called an automorphism. The set G of automorphisms of Z6 is a group. ...
1
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3answers
117 views

What is a group action, and how can we apply it to Sylow theory

I am studying Sylow theorems at the moment, more specifically trying to solve the following problem that I recently posted: Let G be a finite group which has exactly eight Sylow 7 subgroups. Show ...
2
votes
1answer
125 views

If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. Now this is my ...
2
votes
1answer
136 views

Defining action of an elementary abelian 2-group on a vector space.

I have a group $G = \oplus_{\alpha} \mathbb{Z}/2$, where all the direct summands are indexed by the elements of a set (a list in GAP). I want to define in GAP an action of this group on a vector ...
1
vote
1answer
43 views

Proving that the dihedral group $D_n$ has $2n$ elements

I am trying to prove that the dihedral group $D_n$ has $2n$ elements by using the theory of group actions. Specifically I want to use the orbit stabilizer theorem. So I need $D_n$ to act on a specific ...
0
votes
0answers
99 views

Reference for theorem on non-decreasing functions of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose $k\colon M\rightarrow N$ is a function such that $k(\epsilon)=\epsilon$ for all $a,b\in M$, there exists a unique $v\in N$ such ...
5
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2answers
75 views

Difference between Stabilizer and Centralizer?

I know that the Centralizer of an element $a$ in a Group $G$ is defined as follows $$C_G(a) = \{ g \in G \space | \space ga = ag \}$$. It can also be defined as follows $$C_G(a) = \{ g \in G \space ...
2
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0answers
17 views

What are the $SL(2, \mathbb{R})$-invariant Baire measures on $\mathbb{R}^2$?

This was an interesting problem that came up in qual studying. $SL(2, \mathbb{R})$ acts on $\mathbb{R}^2$, and hence on any measure on $\mathbb{R}^2$. What are the Baire measures (i.e. Borel ...
11
votes
3answers
294 views

Elementary Combinatorial Proofs using group action

In trying to prove that the number of spanning trees in $K_5$ is $125$ I adopted the following method: Let $S$ be the set of all such spanning trees and let $S_5$ act in a natural way on $S$. Now ...
2
votes
1answer
91 views

Show that every $A \in SL_3-$action has at least 3 fixed points on $\mathbb{P}^2$.

Consider the natural action of $SL_3(\mathbb{C})$ on $\mathbb{P}^2$ via: $$ SL_3(\mathbb{C}) \times \mathbb{P}^2 \to \mathbb{P}^2, \ (A,[v]\mapsto [Av]). $$ It is clear that the kernel of the ...
1
vote
2answers
191 views

Orbits of properly discontinuous actions

Definition Let $G$ be a group and $X$ a topological space. Let $G\curvearrowright X$ by homeomorphisms. We call the action properly discontinuous if for all $x\in X$ there exists an open neighborhood ...
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0answers
34 views

How can I obtain these differential operators for this transformation?

I have transformation as the following form \begin{eqnarray} \begin{split} &u \longrightarrow \bar{u}=(ax+by+\eta)^{-3} u,\\ &x \longrightarrow \bar{x}=\frac{\alpha x+\beta ...
1
vote
1answer
114 views

Is $BG =EG / G$ a CW complex?

am currently working through J.Rosenbergs construction of classifying Spaces for the +-construction of higher K-Theory. He defines a CW-complex EG as the direct limit of the inductively defined Spaces ...
1
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0answers
16 views

Orbit space of $S^5$ under certain action

Let $Z/6 = <w| w^6 = 1>$.Consider a $Z/6$-action on $S^5$ as generated by the relation $w.(z_1 , z_2 , z_3 ) = (wz_1 , w^2z_2 , w^3z_3)$ ,where $(z_1,z_2,z_3)$ belongs to $S^5$. Question 1: ...
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2answers
47 views

what is the role played by the group in group action?

$H = Z_5 $ and $G = S_5$ and let $A =\{1,2,3,4,5 \} .$ My question is 1.What is difference between the the action of H on A and G on A ? 2.How many group actions are possible in the above a cases ? ...
2
votes
1answer
70 views

ordinary cohomology from equvariant cohomology

Is it possible that the ordinary cohomology of a space can be obtained from its equivariant cohomology? action is algebraic torus action and space is nonsingular complete complex algebraic variety ...
0
votes
1answer
77 views

Prove: There is a $g \in G$ such that $\forall$ $x \in X: g \circ x \neq x$

I have to prove this theorem for my math study: Let $G$ be a finite group, and $X$ a set with #$X \geqslant 2$. Let the action of $G$ on $X$ be transitive. Prove: There is a $g \in G$ such that ...
0
votes
1answer
45 views

computing the orbits for a group action

Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then the number of orbits for the action of $G$ on the fields with nine elements is 3 5 6 9 I have ...