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.

learn more… | top users | synonyms

1
vote
0answers
35 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
23 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
46 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
46 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
32 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
vote
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
83 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
59 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}\...
4
votes
2answers
64 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 ...
5
votes
3answers
93 views

Is the action of $\mathbb Z$ on $\mathbb R$ by translation the only such action?

It is well known that $\mathbb Z$ acts on $\mathbb R$ by translation. That is by $n\cdot r=n+r$. The quotient space of this action is $S^1$. Could someone give me an example where $\mathbb Z$ acts ...
5
votes
1answer
66 views

On lifting an action of $G$ on $X$ to an action of $G'$ on $\tilde{X}$.

I am reading the section on covering actions from Glen Bredon's Tranformation groups. Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path ...
0
votes
1answer
39 views

Symmetries of octahedron with $2$-faces action

Want to do the $2$-faces action. We use the Orbit stabilizer theorem. Let $X$ be the set of faces (any face can go to any face), $X=\{1,2,3,4,5,6,7,8 \}$. Where $1,2,3,4$ are the front faces of the ...
1
vote
0answers
59 views

Group Action highly transitive

$\Omega$ is infinite set, $X$ is a primitive permutation group on $\Omega$, then why if $Alt(\Omega) \leq X$ (that is, $Alt(\Omega)$ is a subgroup of $X$), then $X$ is highly transitive?
1
vote
1answer
26 views

If $H$ fixes three points, then could the normalizer of $H$ induce an orbit of size two on the fixed points

Let $G$ be a transitive permutation group of degree $\ge 5$ acting such that every four-point stabilizer is trivial. Equivalently this means that every nontrivial element has at most three fixed ...
1
vote
1answer
23 views

If $A_6$ acts on set of size $45$, then every involution fixes five points

Let $G \cong A_6$ and suppose that $G$ acts as a transitive permutation group on a set $\Omega$ of size $45$. I want to prove that every involution fixes five points. Any ideas how this could be done? ...
1
vote
0answers
26 views

If $PSL_2(2^k)$ acts on odd set such that $|\mbox{fix}(g)|\le 3$ for $g\ne 1$, then $G_{\alpha}$ has element of order $\frac{q-1}{\gcd(q-1,3)}$

Let $G \cong PSL_2(q)$ where $q = 2^k \ge 8$. Suppose $G$ acts as a transitive permutation group on $\Omega$ such that $|\mbox{fix}(g)| \le 3$ for nontrivial $g \in G$. Assume $|\Omega| = |G : G_{\...
0
votes
0answers
29 views

In permutation group with $3$-group $H := G_{\alpha}$ and $|\mbox{fix}(H)| = 3$ and regular normal subgroup $N$, we have that $C_N(H)$ has even order

Let $G$ be a finite transitive permutation group on $\Omega$. Suppose that $G_{\alpha}$ is a $3$-group and that $|\mbox{fix}_{\Omega}(G_{\alpha})| = 3$ and that $G$ contains a regular normal subgroup, ...
0
votes
0answers
24 views

If $e \in E$ where $E$ is a component of a permutation group, then $e$ has order $2$ or $3$ if $|\mbox{fix}(g)| \le 3$ for all nontrivial $g \in G$

Suppose $G$ is a transitive permutation group such that all four-point stabilizers are trivial, and $G$ has some nontrivial three-point stabilizer. Said differently this means that every nontrivial ...
0
votes
0answers
22 views

In permutation groups with trivial four-point stabilizers we have $N_G(N_G(M_0)) \le N_G(M_0)$ for $M_0 = G_{\alpha}\cap G_{\beta}\cap G_{\gamma}$.

Let $G$ be a finite transitive permutation group on a set $\Omega$ of odd degree. Suppose that the four-point stabilizers are trivial and that some three-point stabilizer is nontrivial; this is ...
0
votes
1answer
16 views

Properly discontinuous action on hyperbolic plane

If we have G acts properly discontinuously on hyperbolic plane $\mathbb H$, then for any point p $\in \mathbb H$, exist neighborhood V s.t. gV$\cap$V =$\emptyset$ iff gp$\neq$p. Given this, can we ...
1
vote
1answer
26 views

CG-homorphism proof. Stuck at the end!

I am trying to work on some questions back from my uni days, and one has gotten the better of me at the moment! Let $G$ be a finite group and $V, W$ finite-dimensional $\mathbb{C}G$-modules. Let $L:...
4
votes
1answer
40 views

Does $X^{C_2} \simeq * \simeq X/{C_2}$ imply $X \simeq *$?

What the title says. Let $C_2$ be the cyclic group of order 2, and $X$ be a topological space with a $C_2$-action (acting continuously) such that both the quotient space $X/{C_2}$ and the subspace of ...
1
vote
0answers
43 views

If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$

The following is an exercise from D. Robinson: A Course in the Theory of Groups. Let $G$ be a $k$-transitive permutation group of degree $n$ which is neither alternating nor symmetric. Assume $k &...
0
votes
0answers
17 views

Why are two transitive actions of a group G equivalent if there exist an automorphism of G swapping two point stabilisers?

Let $G$ be a group acting transitively on two sets $\Omega_{1}$ and $\Omega_{2}$. Also let $w_{i}\in\Omega_{i}$ and suppose there exists $\alpha\in Aut(G)$ such that $\alpha(G_{w_{1}})=G_{w_{2}}$, ...
1
vote
0answers
30 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
1
vote
0answers
42 views

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$ $H$ is the set of fixed points on $A$ $A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, ...
1
vote
1answer
22 views

Prove the set of permuations which permute only finitely many elements is a normal subgroup

Let $A$ be a non-empty set and let $X$ be a subset of $S_A$ Now let $F(X) = \{a \in A : \sigma(a) = a, \forall \sigma \in X\}$, $M(X) = A\backslash F(X)$, and $D = \{ \sigma \in S_A : \mid M(\sigma)\...
6
votes
3answers
68 views

$H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\} , \forall g \in G \setminus H$ , then $|\cup gHg^{-1}|>\dfrac 12 |G|+1$

Let $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\}$ for all $g \in G \setminus H$. Then is it true that $$|\cup_{g \in G \setminus H}gHg^{-1}|>\dfrac 12 |G|+1$$ If ...
1
vote
2answers
56 views

$G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$?

Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing ...
2
votes
1answer
31 views

A description of the transporter $\operatorname{Tran}_G(H,K)$ for subgroups $H\le K$

Let $H$ and $K$ be subgroups of a group $G$. The transporter of $H$ into $K$ is the set of all $g\in G$ that conjugate $H$ into $K$: $$\operatorname{Tran}_G(H,K)=\{g\in G\mid gHg^{-1}\le K\}$$ ...
1
vote
1answer
29 views

Invariant complement to a $G$-module (not necessarily a vector space)

Let $G$ be a group, $R$ a ring (not necessarily a field), and $M$ an $R$-module. Assume we have a group action $\rho:G \times M \to M$. If there exists a $G$-invariant submodule $N \subseteq M$, is ...
0
votes
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 $m.x$ ...
0
votes
1answer
34 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 ...
1
vote
0answers
47 views

General linear group acting on vector space

I have this question really stuck on it Let G denote general linear two by two matrices over field Fp for a prime p acting on the vector space of two column vectors over Fp i cannot find orbits or ...
0
votes
1answer
26 views

Properties of minimal $\mathbb{Z}$-actions on infinite compact spaces

How does one prove that (1) a minimal $\mathbb{Z}$-action on an infinite compact Hausdorff space is free? (2) for such an action, we can find a nonempty open subset $U$ of the space such that $nU\...
0
votes
0answers
26 views

Finding a map from $X = (0,\infty) \times (0,\infty)$ to a cone

Determine the quotientspace $X / \Gamma$, where $\Gamma = <\phi>$, $\phi(x,y) = (x/2,2y)$ and $X = (0,\infty) \times (0,\infty)$. I think the quotient space has to be a cone, but I can't figure ...
0
votes
0answers
44 views

Properly discontinuous action of a group

Let $\Gamma=\{\varphi^n\mid n\in\mathbb{N}\}$ where $\varphi(x,y)=(\frac{x}{2^n},2^ny)$. I am trying to decide if $\Gamma$ defines a properly discontinuous action on $X=(0,\infty)\times(0,\infty)$. I ...
1
vote
2answers
67 views

Is the axiom $g1 = g$ essential for a group action

A group $G$ acts on a set $\Omega$ if (1) $\omega\cdot 1_G = \omega$ (2) $(\omega \cdot g)\cdot h = \omega \cdot (gh)$ for all $\omega \in \Omega$ and $g,h \in G$. But is (1) really essential, ...
1
vote
1answer
35 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: ...
1
vote
1answer
18 views

Smooth actions and stabilizer

Let $G$ be a Lie group acting smoothly on a smooth manifold $M$. We consider a point $x$ of $M$ and $g$ an element of its stabilizer $G_x$. The smooth diffeomorphism $\theta_g$ of $M$ defined by $\...
1
vote
1answer
38 views

$G$ is doubly transitive on a set $S$ if and only if $G=HTH$ where $H$ is an isotropy subgroup and $T$ is a group of order 2 not contained in $H$.

This is Exercise 47(b) from Chapter 1 of Lang's Algebra. In part $(a)$ one is asked to show that $G$ is doubly transitive if and only if $H$ is transitive on $S\setminus\{s\}$. I've been able to ...