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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0answers
25 views

Does the fundamental group ALWAYS act discretely on the universal cover? [on hold]

I need a sharp answer at least in case of manifolds. Please give references for theorems or lemmas that you cite. Thanks.
2
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1answer
306 views

Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
0
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1answer
53 views

Let a group G act on a set $X$, and suppose that $x,y \in X$ lie in the same orbit. Prove that $G_y=g^{-1}G_xg$ for some $g \in G$

Let a group G act on a set $X$, and suppose that $x,y \in X$ lie in the same orbit. Prove that $G_y=g^{-1}G_xg$ for some $g \in G$ Ok, lets assume $G$ acts on $X$,where $x,y \in X, g \in G$ and ...
2
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0answers
37 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
vote
1answer
49 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 ...
1
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0answers
35 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)
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0answers
28 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
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1answer
27 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
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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
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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 = ...
2
votes
1answer
21 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 ...
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 ...
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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?
3
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1answer
29 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
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 ...
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 ...
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 $\mathbb{R}$ ...
5
votes
1answer
58 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
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0answers
47 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 ...
1
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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 ...
2
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1answer
62 views

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$.

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$. I think this is associated with the action of the left coset of ...
4
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1answer
160 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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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
35 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 ...
2
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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 ...
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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 ...
0
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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 ...
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 ...
1
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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). ...
2
votes
2answers
88 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
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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 ...
0
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1answer
33 views

How does cycle index change along an equivariant map?

Question. Suppose $G$ acts on $X$ (via $\Psi$) and on $Y$ (via $\Phi$), and let $f : X\to Y$ be an equivariant map ($f(g\cdot x) = g\cdot f(x)$ for all $x$ in $X$ and $g$ in $G$). Is there a formula ...
0
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1answer
58 views

Group Action: Group $(\mathbb{Z}, +)$ acting on $\mathbb{R}$

In my group theory notes I have the following: The Group $(\mathbb{Z}, +)$ acts on $\mathbb{R}$ as follows: $m\in \mathbb{Z}$ and $r\in\mathbb{R}$: $m.x \to (-1)^mr$ in this notation $m.x$ ...
1
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2answers
50 views

Question about action on groups in Bourbaki (Algebra I)

In Bourbaki, Algebra I, chapter I, §5 "Groups operating on a set" paragraph 1, Bourbaki defines the operation of a group $G$ on a set $E$ as a morphism $\alpha \in G\mapsto f_\alpha \in S(E)$ ($S(E)$ ...
4
votes
0answers
53 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 ...
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; ...
1
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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
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0answers
38 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 ...
1
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 $\mathbb{R}^{N}$ ...
3
votes
1answer
73 views

An example of lifting a group action to the universal cover.

Through a previous question, I understood how we can lift the action of a group $G$ on a topological space $X$ to an action of a covering group $G'$ of $G$ on the universal cover $\tilde{X}$ in such 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 ...
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 ...
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...