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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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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7 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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24 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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3answers
58 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 ...
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1answer
21 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 ...
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0answers
33 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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2answers
35 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 ...
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1answer
27 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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13 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 ($f\circ\Psi = \Phi$). Is there a formula relating the cycle indices $Z_\Psi$ and ...
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1answer
22 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 ...
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45 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 $m.x$ ...
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0answers
27 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 ...
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1answer
16 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 ...
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24 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 ...
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38 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 ...
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2answers
63 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
16 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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0answers
13 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
30 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
57 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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Do we gain anything interesting if the stabilizer subgroup of a point is normal?

Let $G$ be a group and $S$ a $G$-set with action $(g,s) \mapsto gs$. For some $s \in S$, let the stabilizer of $s$, $G_s=\{g \in G\,|\,gs=s\}$ be normal in $G$. What does this let us say about the ...
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1answer
43 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 ...
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0answers
24 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 $G_{\alpha} \cap G_{\beta}\cap G_{\gamma} < G_{\alpha}$ ...
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1answer
53 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
57 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 ...
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1answer
59 views

If $N_G(U) = TU$ for $T = \langle t \rangle$ with involution $t$, and $N \cap U \ne 1$, then $G = TUN$ and $UN$ is Frobenius group

Let $U \le G$ be a subgroup of the finite group $G$ of odd order such that $|N_G(U) : U| = 2$ and different conjugates of $U$ intersect trivially, i.e. $U^g \cap U = 1$ for $g \notin N_G(U)$. Suppose ...
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1answer
23 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 ...
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3answers
37 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
39 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 $E$. ...
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vote
1answer
28 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 ...
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0answers
29 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 ...
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1answer
23 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 ...
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votes
1answer
35 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
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1answer
34 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 ...
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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 ...
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1answer
49 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 $\phi_g$ ...
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33 views

If $G$ has cyclic Sylow $2$-subgroups, then the core $O(G)$ acts transitive.

Let $G$ be a finite, transitive permutation group on $\Omega$, and assume the point stabilizers have even order. Denote by $O(G)$ the largest normal subgroup of $G$ whose order is odd (see here for ...
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0answers
32 views

Notions of groups acting on groups

Let $G$ be a group acting on a set $S$, by means of $(g,s)\mapsto s^g$. If $S$ is itself also a group, then it is natural to impose the further condition that $(st)^g=s^gt^g$. This seems to be the ...
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0answers
47 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,3\}$ for $g \ne 1$, and stabilizers are t.i. subgroups, then the Sylow $3$-subgroups have maximal class

Let $G$ be a transitive permutation group such that every nontrivial element fixing some point fixes exactly $3$ points. Also assume that for $g \notin N_G(G_{\alpha})$ we have $$ G_{\alpha} \cap ...
4
votes
1answer
59 views

How are $G$-modules and linear group actions different

Let $M$ be an abelian group and let $G$ be a group acting on $M$ such that $M$ is a $G$-operator group, i.e. we have for $u, v \in M$ and $g,h \in G$ (1) $u\cdot 1_G = u$ (2) $(ug)h = u(gh)$ (3) ...
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2answers
47 views

Stabilizer, Kernel and Orbit of the right action.

If $G$ act on the set of all right cosets of a subgroup $K$ then I have the following questions:- What is the stabilizer of an element $Kx$. What is the kernel of the Action. What is the orbit of ...
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1answer
56 views

Classify orbits of conjugating action on $GL_2(\mathbb{C})$

We have a general linear group $GL_{2}(\mathbb{C})$ that acts on the set of $M_2(\mathbb{C})$ set of $2\times 2$ matrices by conjugation. I want to classify the orbits of its action. What I know:- ...
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0answers
19 views

Mobius transformation and Group action

Let $G$ be $SL_2(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant $1$, acting on $\mathbb{C}\cup \infty$ by M¨obius transformations. For each of the points $0$, $i$, $−i$, ...
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0answers
23 views

Group Action and centraliser

Let Sn be the group of permutations of {1,...,n}, and suppose n is even, n > 4. Let g = (12) ∈ Sn, and h = (12)(34)... (n−1 n)∈ Sn. (i) Compute the centraliser of g, and the orders of the ...
0
votes
1answer
28 views

Let $G = PSL(2,q)$ with $q$ odd and $H$ be a subgroup of even order. Then the centralizer of an involution in $H$ is a dihedral group.

Let $G = PSL(2, q)$ with $q = p^n$ and $p \ne 2$. Suppose $H$ is a subgroup of even order. Then $H$ contains an involution $u$. Assume that $N_G(H)$ contains the centralizer of $u$. The ...
1
vote
0answers
29 views

Group actions on a set

Let $G$ be a finite group acting transitively on the set $A$. Let $p$ be a prime and $S \in Syl_{p}(G)$. Show that $N_{G}(S)$ acts transitively on $Fix_{A}(S)$(the set of fixed-points of S in A). I ...
2
votes
3answers
183 views

If G acts on X, show that there must be a fixed point for this action. Please help. [closed]

Suppose that $G$ is a group of order $p^k$, where $p$ is prime and $k$ is a positive integer. Suppose that $X$ is a finite set and assume that $p$ does not divide the $|X|$. If $G$ acts on $X$, show ...
5
votes
1answer
59 views

The closure of $\mathbb{Z\times Z\times R}$ in Homeo$(\mathbb R^2)$ is the group of translations of $\mathbb R^2$

This question is related to this question but not the same - Let $\mathbb{Z\times Z\times R}$ act on $\mathbb R^2$ by $$(m,n,r)\cdot(x,y)=(x+m+r,y+n+r\sqrt{2})$$ I need to prove the following - If ...
0
votes
1answer
39 views

$G$-graded vector space and module

Let $G$ be a finite group and let $k(G)$ be the set of functions on $G$ with values in a field in $k$. I am reading a proof of the following fact: a $k(G)$ module is a $G$-graded vector space $V$. ...
2
votes
0answers
32 views

How to describe the points of a quotient stack?

Let $G$ be a finite algebraic group acting on a projective complex variety $X$. Then a quotient $Y=X/G$ exists as a scheme and, if $G$ acts freely, $Y$ is an orbit space and the natural map ...