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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4
votes
1answer
151 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)
0
votes
2answers
18 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
28 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
24 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 ...
0
votes
0answers
28 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
votes
0answers
32 views

$G$ has size $55$ and acts on a set of size $39$ [closed]

Let $G$ be a group with size $55$ which acts on a set with size $39$. Prove that this action has a fixed point (which means there exists a $x$ in the set such that $\forall g\in G : gx=x$.) I don't ...
2
votes
0answers
22 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 ...
-1
votes
1answer
152 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
votes
1answer
18 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
23 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
vote
0answers
14 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
76 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
47 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
votes
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
votes
1answer
57 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
vote
2answers
47 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
51 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
60 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
vote
0answers
25 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
33 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
33 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
29 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
12 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
71 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
72 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
24 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
38 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 ...
1
vote
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...
0
votes
0answers
11 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 ...
1
vote
1answer
32 views

Questions about the definition of a periodic pattern

In this article, Doris Schattschneider defines what a repeating (or periodic) pattern is. The definition goes as follows: A periodic pattern in the plane is a design having the following property: ...
3
votes
0answers
55 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\ \ :\ \ ...
4
votes
2answers
54 views

Group Operations/ Group Actions

I'm currently taking my first abstract algebra course and am learning about group actions, orbits, and stabilizers. I'm reading the Artin textbook and I am not very clear of what exactly a group ...
0
votes
3answers
112 views

Examples of a map involving group actions

Okay, this is a trivial question but I need some non-trivial examples of a map involving group actions. What I mean: Let $G$ be a group acting on a set $A$. Let $G'$ be another group acting on ...
2
votes
3answers
178 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 ...
3
votes
3answers
491 views

Concrete examples of group actions.

First, a little motivation: I have read the section on Group Actions in Dummit & Foote, the wikipedia page, and (countably many) other references. And seemingly without exception, they only offer ...
4
votes
1answer
42 views

If $G$ acts primitively and $\Gamma \subseteq \Omega$ is not a block, then each pair of points could be separated

Let $G$ act on $\Omega$. A subset $\Delta \subseteq \Omega$ is called a block if for each $x \in G$ either $\Delta^x \cap \Delta = \emptyset$ or $\Delta^x = \Delta$, where $\Delta^x := \{ \delta^x : ...
1
vote
1answer
29 views

Is nonsingular group action on a measure space “continuous”?

Suppose $G$ is a group acting on a nonatomic standard measure space $(X,\mu)$ (say $[0,1]$ with Lebesgue measure). Assume that the action is nonsingular, i.e. $\mu(E)=0$ implies $\mu(gE)=0$ for all ...
1
vote
4answers
60 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 ...
2
votes
2answers
34 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 ...
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
42 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
26 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 ...
5
votes
0answers
419 views

Semidirect product group actions

$H$ and $K$ are groups, and $\Gamma$ is a set acted upon by H, while $\Delta$ is a set acted upon by $K$. Let $W := K \wr_\Gamma H$, the wreath product of $H$ and $K$. I have seen theorems stating ...
4
votes
1answer
56 views

Show $g.S = \{g.x|x\in S\}$ defines an action on the power set

Let $X$ be a finite set and let $\mathscr{P}(X)$ be its power set. A group $G$ acts on $X$. Given $g \in G$ and $S \subseteq X$ show $g.S = \{g.x|x\in S\}$ defines an action on $\mathscr{P}(X)$. I ...
1
vote
0answers
33 views

Group action on subspaces of $\mathbb{R}^4$

Let $V=\mathbb{R}^4$. Let $S$ be the set of all two-dimensional subspaces of $V$ and fix $W\in S$. Let $G=GL(V)$ (the group of invertible linear operators on $V$) act naturally on $S$ and let ...
0
votes
0answers
21 views

Does a function between sets induce a homomorphism between the respective permutation groups?

Let $X,Y$ be finite sets, and let $\Sigma(X),\Sigma(Y)$ be their respective permutation groups. Consider a function $f:X\to Y$. Is there a homomorphism $\phi:\Sigma(X)\to\Sigma(Y)$ induced ...
3
votes
1answer
40 views

Where can I found an explanation of group cohomology from the point of view of invariants?

I heard once that we can view group cohomology as the right derived functor quantifying precisely (i.e. by the usual long exact sequence) how much the functor of "taking the invariants" is not right ...