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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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
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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}\...
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2answers
63 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
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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
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1answer
63 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
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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 ...
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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?
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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 ...
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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? ...
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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_{\...
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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, ...
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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 ...
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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 ...
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1answer
15 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
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1answer
23 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 ...
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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 &...
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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 $\alpha(G_{w_{1}})=G_{w_{2}}$, ...
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0answers
29 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 $\mathbb{F}(V)$ ...
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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, ...
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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
51 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
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1answer
29 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 $K$: $$\operatorname{Tran}_G(H,K)=\{g\in G\mid gHg^{-1}\le K\}$$ ...
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1answer
27 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 ...
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0answers
46 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
24 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
42 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, ...
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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: ...
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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 $\...
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1answer
37 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 ...
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1answer
62 views

Every group of order $60$ , having a normal subgroup of order $2$ , has a normal subgroup of order $6$ (without Sylow )?

How to prove , without using Sylow's theorems , that every group of order $60$ , having a normal subgroup of order $2$ , contains a normal subgroup of order $6$ ? Please help . Thanks in advance
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27 views

Let $G$ be a permutation group and $R \unlhd G$. If $G$ acts double-transitive on the orbits of $R$, then $G / R \cong A_5$ and we have $5$ $R$-orbits

Let $G$ be a transitive permutation group such that every nontrivial element fixing some point fixes exactly three points. Also suppose that $G_{\alpha} \cap G_{\beta}\cap G_{\gamma} < G_{\alpha}$ ...
0
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1answer
62 views

More question on the proof of orbit-stabilizer theorem from Gowers's weblog

Still I'm reading Gowers's weblog about orbit-stabilizer theorem, I must admit that my understanding of this materiel improved, but still I have some question. Let $G$ be a finite group, and $X$ be ...
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1answer
59 views

Show that $|C_G(u)| = 12$ by “counting involutions”

Let $G$ be a transitive permutation group acting on $\Omega$ such that every non-trivial element fixing some point has exactly three fixed points. Suppose $G_{\alpha} \cong A_5$ for some point ...
0
votes
1answer
26 views

If $G = VN$, $V$ a four group and $N$ regular normal, then there exists some Sylow subgroup left invariant by $V$

Let $G$ be a permutation group on $\Omega$ with $G = VN$, where $V \cong C_2 \times C_2$ (the four-group) and $N$ has odd order with some prime divisor $>3$. Suppose $N$ is a regular normal ...
0
votes
3answers
42 views

Groups of prime power and the fixed point set

Suppose that $X$ is a finite $G$-set. A group $G$ is of prime power if $|G|=p^n$ for $p$ prime. The fixed point set $X_G=\{x\in X : gx=x$ $\forall g\in G\}$. I'm asked to prove that $|X|=|X_G|$ (mod ...
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2answers
42 views

A subtle error in a “change of variable” in a sum

Let $G$ be a finite group of order $n$, and let $E$ be a finite set. Let $\star$ be an action of $G$ on $E$. Suppose that $G \star x_1,..., G \star x_m$ are the distinct orbits of elements in $E$. ...
0
votes
1answer
44 views

If $G_{\alpha} \cong S_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$. Then $G$ has transitive normal subgroup of index $2$.

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
2
votes
1answer
42 views

Significance of the notion of equivalent actions vs. permutation isomorphic action

Let $G$ be a group acting on $\Delta$, and $H$ be a group acting on $\Gamma$. If there exists an isomorphism $\varphi : G \to H$ and a bijection $\psi : \Delta \to \Gamma$ such that $$ \psi( \alpha^g ...
0
votes
0answers
32 views

Two actions that should be non-equivalent on $A_4$, but they seem to be equivalent?

I was trying to find some actions that are permutation isomorphic, but not equivalent. See my recent post here for the definitions. One natural candidate seems $A_4$. As the subgroups $U_1 = \langle (...
1
vote
1answer
29 views

Example in which a normal subgroup acts non-equivalent on its orbits

Let $G$ be a group acting on $\Delta$, and $H$ be a group acting on $\Gamma$. If there exists an isomorphism $\varphi : G \to H$ and a bijection $\psi : \Delta \to \Gamma$ such that $$ \psi( \alpha^g ...
2
votes
1answer
42 views

Given $\pi:X\rightarrow Y$ how to show $X$ is irreducible (resp. normal) $\Rightarrow$ $Y$ is irreducible(resp. normal)?

Let $G$ act on the affine variety $X=\operatorname{Spec}(R)$ such that $R^G$ is a finitely generated $\mathbb C$ - algebrs and let $\pi:X\rightarrow Y=\operatorname{Spec}(R^G)$ be the morphism of ...
2
votes
1answer
50 views

Transitive action on a finite set and group

If $G$ is a finite group and acting transitively on a set $X$ with $|X|>1$. then I have two question :- There is some element of $G$ in which fixes no element of $X$. Give a counter-example to ...
0
votes
0answers
23 views

If $G_{\alpha} \cong A_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$ and $V \le G_{\alpha}$ is the four-group in $A_4$, then $C_G(V) = V$

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
3
votes
1answer
56 views

Why is this a group action - what is the significance of $g^{-1}$?

Let $G$ be a group acting on a variety $X$ such that every $g\in G$ defines a morphism $\phi_g:X\rightarrow X$ given by $\phi_g(x)=g\cdot x$. If $X=\operatorname{Spec}(R)$ is affine then $\phi_g$ ...