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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2
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0answers
22 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
1
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0answers
21 views

Let $G$ be a $p$-group, with $|G|= p^n$. Show that $G$ has a normal subgroup of order $p^m$ for each integer $0 < m < n$. [duplicate]

I think I have solved a problem using one of the sylow theorems. But, if this proof is correct, I think I've cheated a little. Since the chapter on Sylow theorems comes directly after the chapter on ...
-1
votes
0answers
27 views

Self conjugation on a group with $2p$ elements

Let $G$ be a group with $2p$ elements where $p$ is an odd prime. Also let $Z(G)=e$, where $e$ is the identity element. Prove that there is a conjugationclass with $p$ elements. My attempt: Because ...
1
vote
1answer
26 views

Non-zero fixed point of some linear action on any finite group

Let $G$ be a group , $F$ be a field , $n$ be a positive integer , a map $h:G \times F^n \to F^n$ is called a linear action if there is a group homomorphism $f:G \to GL(n,F)$ such that ...
1
vote
1answer
41 views

$GL_2(\Bbb C)$ acts on a certain set

Let $G:=GL_2(\Bbb C)$, $B$ and $T$ be the subgroup consisting of all upper triangular and diagonal matrices in $G$, respectively. Set $w:= \left( \begin{array}{cc} 0 & 1\\ 1 & 0 ...
1
vote
2answers
46 views

The orbit of a compact Lie group action

Let $G$ be a compact Lie group acting on a manifold $M$. For each $p\in M$, we define the orbit of $p$ as $G\cdot p:=\{g\cdot p: g\in G\}$. The isotropy group of $p$ is $G_{p}=\{g\in G:g\cdot p = ...
1
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0answers
31 views

Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
2
votes
1answer
18 views

a basic question in crossed product for compact group action

I am quite new into crosssed product of Fréchet algebras or C$^*$-algebras. So if the question is too basic please excuse me. Suppose we have two Fréchet algebras or C$^*$-algebras $A$ and $B$ and ...
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0answers
10 views

Representatives of the conjugacy classes of s5 [duplicate]

List the partitions of 5 and corresponding representatives of conjugacy classes in s5. What is the procedure to find the representatives of the conjugacy classes?
0
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0answers
16 views

Action of a Galois group on a discrete module is continuous if every stabilizer has finite index.

Let $G=G(\overline{K}/K)$ be the Galois group of the separable closure $\overline{K}/K$. And Let $M$ ($M$ has discrete topology) be an abelian group on which $G$ acts additively, i.e., $\sigma \circ ...
3
votes
1answer
26 views

Example of a free group action that is not proper.

I have been trying to think about Lie group actions on smooth manifolds and what the quotient spaces look like. I have a proof that compact Lie groups produce proper actions on manifolds, as well as ...
1
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0answers
82 views

Dense and turbulent orbits

In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition ...
1
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1answer
18 views

Subsets of $G$-sets with sharply transitive $G$-action

Let $G$ be an infinite group acting sharply transitively on a set $X$. Let $Y\subset X$ be a proper subset. Is there a subgroup $H\leq G$ which acts sharply transitively on $Y$ ? I think this is ...
1
vote
0answers
18 views

Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description

What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ...
5
votes
1answer
56 views

A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n $

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
0
votes
0answers
46 views

Show that $S^1$ acts on $S^3$

$S^3=\{(z_1, z_2) \in \mathbb{C^2} \mid |z_1|^2 + |z_2|^2 = 1 \}$ Show that $S^1$ acts on $S^3$ by $z \cdot (z_1, z_2)=(zz_1, zz_2)$ An action of a topological group $G$ on a topological ...
5
votes
1answer
62 views

Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ ...
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0answers
16 views

circle actions on spheres

I'm considering the following action of $S^1$ on $S^3$: $$ e^{i\theta}.(z_1,z_2)=(e^{i\theta}z_1,e^{iq\theta}z_2) $$ It is clear that when $q=1$ the quotient space is $S^2$. Is there any description ...
0
votes
2answers
20 views

Group action decomposes $X$ into distinct orbits

Define the group action as $g\cdot x:=g^{-1}xg.$ Let $G=A_5$, and $X=\{\sigma\in A_5:=\sigma=(a,b,c,d,e)\}.$ Show that the group action on X decomposes $X$ into two distinct orbits. There are 60 ...
0
votes
1answer
34 views

Group action and equivalence relation

Let $G$ be finite, and group action on $X\subseteq G$: $g\cdot x:=g^{-1}xg$. Let $G=S_n$, and $X=S_n.$ Show that $[x]_R$ consists of all elements of $S_n$ that are of the same cycle-type as $x$. I ...
0
votes
2answers
27 views

Finding the orbits of the orthogonal group $O(n)$ on $\Bbb R^n$

Let $O(n)=\{M\in GL_n(\mathbb{R}):MM^t=M^tM=I\}$ an orthogonal group. I need please an explain why each orbits consists of all vectors with the same length. I know that an orbit is defined by ...
1
vote
1answer
48 views

Natural action of $\mathbb{Z}G$ on $\mathbb{Z}$?

I'm studying projective modules and I'm having problem coming up with (or understanding) examples of non-free projective modules. I got that when a ring is a direct sum $R = A \oplus B$, both $A$ and ...
2
votes
0answers
24 views

Invariant cohomology for non-compact groups

Suppose I have a compact $G$-space $M$, and a differential form $\omega$ on $M$ with the property that $$ \forall g\in G\quad g\omega = \omega + d\lambda_g, \quad(*) $$ i.e. $g\omega$ is cohomologous ...
0
votes
1answer
24 views

Can we discribe a Lie group action from some local property?

Let G be a Lie group,and it acts on a smooth manifold M.Then can we get that the action is transitive from some local property of the Lie group action.More precisely,Can we get the action is ...
1
vote
0answers
15 views

Is this covering group $G'$ of $G$ unique?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
0
votes
1answer
25 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
2
votes
2answers
87 views

Is the fixed point set of an action a submanifold?

Let $M$ be a differentiable manifold, and $G$ a Lie group acting smoothly on $M$. Under which condition - if any - is the set of fixed points of the action a submanifold of $M$? My thoughts so far: ...
1
vote
1answer
50 views

Induction backwards to prove Sylow's first theorem

Claim: Suppose $H\le G$ and $P$ is a Sylow $p$-subgroup of $G$. Show that, without reference to Sylow's theorems, there exists some conjugate of $P$ whose intersection with $H$ is a Sylow ...
-1
votes
1answer
155 views

Group acting on a Projective Space

Let $G$ be an algebraic (zariski closed) subgroup of $SL(n,C)$ for some algebraically closed field $C$. Now $G$ acts on an $n$-dimensional vector space $V$ over $C$ where $V$ is a solution space of a ...
4
votes
0answers
52 views

group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
1
vote
0answers
26 views

Transitive action of a $p$-group on minimal block systems

I have trouble proving the following theorem: Let $P$ be a transitive $p$-subgroup of ${\rm Sym}(A)$ with $|A| > 1$. Then any minimal $P$-block system consists of exactly $p$ blocks. Furthermore, ...
2
votes
0answers
37 views

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.
2
votes
1answer
36 views

On the action of galois groups in towers of fields

I would like some confirmation on certain statements I believe to be true: Let $K\subset L\subset M$ be a tower of fields such that both extensions $L/K$ and $M/K$ are galois. Let $f(x) \in K[x]$ be ...
2
votes
1answer
30 views

Classify orbit of $G=GL_2(\mathbb{C}) \times GL_2(\mathbb{C})$ acts on the set $M_2(\mathbb{C})$

The group $G=GL_2(\mathbb{C}) \times GL_2(\mathbb{C})$ acts on the set $M_2(\mathbb{C})$ of $2\times 2$ matrices as follows:- $(f,g)(x)=fxg^{-1}, f,g \in GL_2(\mathbb{C}), x\in M_2(\mathbb{C})$. I ...
0
votes
1answer
21 views

$G$ is a group of order $12$ admitting an irreducible $3-$dimensional reprsentaion. What are the dimensions of its irreducible representaions?

Given $G$ is a group of order $12$ admitting an irreducible $3-$dimensional representaion. What are the dimensions of its irreducible representaions? Is there a theorem that gives an answer? I am ...
4
votes
1answer
63 views

Can every monoid action be turned into a group action?

Let $\mathbf{Mon}$ be the category of monoids. Let $\mathbf{Grp}$ be the category of groups. There is the inclusion functor $i : \mathbf{Grp} \to \mathbf{Mon}$. It has both a left and a right adjoint; ...
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vote
0answers
13 views

$\mathbb{R}^{N}/\Sigma_{n}$ as a topological space

Let $\Sigma_{n}$ denote the symmetric group on $n$ letters. $\Sigma_{n}$ acts on unordered pairs $\{i,j\}$ via $\sigma(i,j)=\{\sigma(i),\sigma(j)\}$. Let $e_{\{i,j\}}$ be a basis for $\mathbb{R}^{N}$ ...
1
vote
0answers
27 views

Descent of line bundles

If a finite group acts $G$ on a variety $X$, consider the quotient $X/G$. I would like to understand which line bundle on $X$ descends to $X/G$. The action is not free. Can anyone direct me to some ...
1
vote
0answers
40 views

The action of a topological group on the function space is continuous?

Sorry for my bad english. Let $X$ and $Y$ be two topological spaces, and $G$ a topological group, let $\theta : G \times X \to X$ be a continuous action of $G$ on $X$. We defined the action of $G$ on ...
1
vote
1answer
24 views

Irreducible rep, group centre: $\pi$(z) $=\lambda$(z)v

Note: not sure if title is displaying well; formula is directly below lambda is a scalar that I need to show exists $\pi$(z) $=\lambda$(z)v lambda is a scalar that I need to show exists I want to ...
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votes
0answers
12 views

Question on subgroup cohomology restricting proper, simplicial actions of an algebraic group

I have a question regarding an assertion made in p. 2 of these notes on Bruhat-Tits buildings. The question concerns the group $G_p=SL_n(\mathbb{Q}_p)$ and its subgroup $\mathbb{Z}^{n-1}$ (the ...
0
votes
2answers
21 views

In a finite p-group,H is a maximal sub group iff H is normal in G and |G:H|=p

Let G be a finite p-group,H is a maximal subgroup of G if and only if H is normal in G and |G:H|=p I tried acting H on right cosets of H in G .... I don't know what to do now...
2
votes
2answers
35 views

How to show that $SL_2(\Bbb R)/SO_2(\Bbb R) \cong \Bbb H$?

I've already shown that $SL_2(\Bbb R)$ acts on $\Bbb H$ on the left : $$SL_2(\Bbb R) \times \Bbb H \rightarrow \Bbb H$$ $$\gamma*z \mapsto \frac{az + b}{cz + d}$$ where $\gamma = \begin{pmatrix} a ...
3
votes
1answer
73 views

Homorphism from $B(G)$ to $\mathbb{Z}$

Let $G$ be a finite group, and $B(G)$ be its Burnside ring. Show that each ring homorphism $\varphi:B(G)\to\mathbb{Z}$ is the mark of some $H\le G$, i.e. it maps to an equivalent class of finite ...
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vote
4answers
62 views

Definition of quotient of a topological space by a group action

I was going through the following lecture note on topology as I was trying to understand quotient topology . http://homepage.math.uiowa.edu/~jsimon/COURSES/M132Fall07/M132Fall07_QuotientSpaces.pdf ...
2
votes
1answer
16 views

size of group of row and column flips of a square board

Let $X$ be the set of numberings of the squares in a $n \times n$ board with the numbers $1$ to $n^2$. Let $G$ be the group of transformations of boards generated by row and column flips, where a flip ...
3
votes
0answers
56 views

Show that $ \mathcal{D}_H:=\bigcup_{g_i\in[H\backslash G]} g_i\cdot \mathcal{D} $ is a fundamental domain

Let $G$ be a group which acts on the set $X$. Consider a subgroup $H$ of $G$ which acts on $X$ by the restriction of the action of $G$ on $X$. Let $[H\backslash G]:=\{g_i\ \ :\ \ ...
0
votes
0answers
14 views

Find $\gamma$ such that $\gamma\cdot z\in\mathcal{D}_{\text{SL}_2(\mathbb{Z})}$

Let $\text{SL}_2(\mathbb{Z}):=\{A\in\mathbb{Z}^{2\times2}\space | \space\det(A)=1\}$ and consider the action of this group on $\mathbb{H}:=\{z\in\mathbb{C}\space | \space \Im(z)>0\}$ defined by: ...
0
votes
0answers
43 views

Exercise of Rick Miranda is wrong? Actions over Riemann sphere

I'm studying the book Rick Miranda, Algebraic Curves and Riemann Surfaces and I have a question about the exercise H of page 84. The book says that $z \mapsto exp(2\pi i /r)z$ is an automorphism of ...
2
votes
0answers
27 views

Dihedral groups acting on Riemann surfaces

I'm studying the quotient riemann surface $X/G$. I'm looking for examples of dihedral groups $D_n$ acting on some riemann surfaces $X$ or at least acting on it's Jacobian JX. Does anybody knows some ...