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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0
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2answers
22 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
36 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
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2answers
31 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 $$G\...
1
vote
1answer
52 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
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0answers
27 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
17 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
26 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
106 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
51 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 $p$-subgroup ...
-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
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0answers
55 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
27 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
42 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
37 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
31 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
65 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
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
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0answers
31 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
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0answers
41 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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0answers
14 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
37 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
74 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 $G$-...
1
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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
72 views
+100

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\ \ :\ \ \exists!\mathcal{O}\...
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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
45 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
28 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 ...
4
votes
1answer
57 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
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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 $H=\{g\...
0
votes
0answers
22 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
43 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 ...
1
vote
1answer
45 views

Does a group action always induce a quotient?

Let $G$ be a group, let $X$ be a set on which $G$ acts (possibly non-faithfully). I would be tempted to say the following: There exists a normal subgroup $K$ of $G$, such that: $G/K$ acts ...
2
votes
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 $H$...
1
vote
1answer
30 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 $g\...
1
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0answers
31 views

If $G = R \rtimes G_{\alpha}$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \ncong D_{12}$

Suppose that $G = R \rtimes G_{\alpha}$ is a finite permutation group on $\Omega$ where $R$ is the non-abelian subgroup of order $27$ and exponent $3$ (see here). Suppose that $G_{\alpha} \cong D_n$, ...
3
votes
1answer
79 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
38 views

Example of a permutation group with fixed point restrictions and dihedral or semidihedral Sylow $2$-subgroups

I am looking for a finite, transitive and nonregular permutation group $G$ acting on $\Omega$, such that every nontrivial element fixes at most two points and such that i) the point stabilizers $G_{\...
0
votes
1answer
48 views

If $|\alpha^N| = 1 + k|g|$ for $g \in G_{\alpha}$, then $g$ fixes a point on every $N$-orbit that it stabilizers

Let $G$ be a finite transitive permutation group, suppose $p$ does not divide $G_{\alpha}$ and that $1 \ne P \unlhd G$ is a normal $p$-subgroup of $G$. Let $\Delta := \alpha^P$ be an orbit of $P$ and ...
5
votes
1answer
56 views

Is the quotient $X/G$ homeomorphic to $\tilde{X}/G'$?

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
0answers
26 views

If $E := F^{\ast}(G) \cong Sz(q)$, then the elements from $G \setminus E$ act as field automorphisms of odd order on $E$

Let $G$ be a finite, transitive, nonregular permutation group on $\Omega$ such that every nontrivial element fixes at most two points. Let $E := F^{\ast}(G)$ be the generalised Fitting subgroup. ...
1
vote
2answers
27 views

What are the fixed points of this action?

For a fixed integer $d$, let $G=\left\{\left(t,\dfrac{1}{t}\right)\in (\mathbb C^*)^2:t^d=1\right\}$ act on $\mathbb C^2$ by ponitwise multiplication. That is $$\left(t,\dfrac{1}{t}\right)\cdot(x,y)=\...
3
votes
2answers
30 views

Action of $A_n$ on cosets by translation

This exercise is from Lang's Algebra. Let $n\geq 3$, and let $H$ be a subgroup of the alternating group $A_n$. Suppose that $H$ has index $n$ in $A_n$. Show that the action of $A_n$ on $A_n/H$ by ...
4
votes
2answers
60 views

Does $\pi_1(X, x_0)$ act on $\tilde{X}$?

Let $X$ be a path connected, locally path connected space and let $p:\tilde{X} \to X$ be a covering map. Let $x_0\in X$. Then we have a natural right action of $\pi_1(X, x_0)$ on the fibre $p^{-1}(x_0)...
2
votes
0answers
34 views

A specific question on three statements in a paper about fixed point bounded groups, its interpretation and its usage w.r.t. the Sylow theorems

This is a rather specific post, but I hope nevertheless someone can help me. I am refering to a specific paper, namely K. Mayaard, R. Waldecker, Transitive permutation groups where nontrivial ...
2
votes
0answers
24 views

If stabilizer contains Sylow $2$-subgroup $S$ and another nontrivial subgroup $X$ fixing two points, then $X$ normalizes $S$

Let $G$ be a transitive, nonregular permutation group acting on $\Omega$. Suppose that $|\Omega|$ is odd, then $G_{\alpha}$ contains a full Sylow $2$-subgroup $S$ of $G$. Suppose that $G = G_{\alpha}\...