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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$G$ 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
61 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 $G_{\alpha} \cap G_{\beta}\cap G_{\gamma} < G_{\alpha}$ ...
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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 ...
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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 ...
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41 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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41 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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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 ...
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1answer
36 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
31 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
28 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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1answer
40 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
48 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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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
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1answer
53 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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1answer
61 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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0answers
37 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 ...
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1answer
70 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
72 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
70 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
40 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
41 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 ...
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31 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 ...
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1answer
42 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 ...
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1answer
47 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 $V$. ...
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45 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 ...
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3answers
55 views

For a subgroup $H$ of a group $G$, how many cosets are there?

According to my lecture notes, for a subgroup $H$ of a group $G$, the (right) cosets of $H$ in $G$ are all the sets given by $$ Hx = \{hx: h \in H\} $$ Where $x \in G$. This implies that the number ...
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1answer
61 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 ...
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1answer
54 views

In what sense is this action of $\mathbb R$ on $T$ lifted to an action of $\pi_1(T)\times\mathbb R$ on $\mathbb R^2$?

I am reading the paper "Calculating the fundamental group of an orbit space" by M A Armstrong where he states the following - Let $\mathbb R$ act on the torus $T\cong S^1\times S^1$ by ...
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0answers
64 views

Group actions by semi-direct products of groups

I have trouble to understand the second part of the following example which I hope someone can explain to me. First let me explain the initial situation which I feel comfortable with: Consider the ...
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1answer
32 views

If $A$ is $G$ - invariant then so is $\bar A$

Let $G$ be a topological group acting continuously on a topological space $X$. Let $A$ be a $G$ - invariant subspace of $X$. Then is it true that $\bar A$ is also $G$ - invariant? (where $\bar A$ is ...
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2answers
47 views

kernel of action.

If $\phi : G \to Perm(G/H)$ where $\phi$ is the group action on $G/H$ by $G$. $\phi := g(g'H) = (gg')H$ Why is the kernel of $\phi$ equal to $\cap_{x\in G} xHx^{-1}$ I thought the kernel is $\ker ...
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34 views

If $G$ acts so that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$. Conditions such $S \in \mbox{Syl}_p(G)$ has maximal class.

Let $G$ be a nonregular, transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Further suppose that for $g ...
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17 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ maximal, why is $N_M(M_{\alpha}) \in \mbox{Syl}_p(M)$?

Let $p$ be an odd prime. Suppose $G$ is solvable and acts as a nonregular and transitive permutation group on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points. ...
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36 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, $M$ maximal with $|G : M| = p$, then $|M / L| = p$ for semiregular $L \unlhd G$.

Let $G$ be a solvable, nonregular and transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. And suppose that ...
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19 views

The uniqueness of the Frobenius representation as handled in textbooks, for example Kurzweil/Stellmacher

As written on Wikipedia:Frobenius_group The Frobenius kernel $K$ is uniquely determined by $G$ as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by ...
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38 views

Group of rotational symmetries of regular tetrahedron is isomorphic to $A_4$

Let $G$ be the group of rotational symmetries of a regular tetrahedron. I'm trying to think of an argument proving that since $G\cong H$, where $H\leq S_4$, $H=A_4$. There are 12 rotational ...
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135 views

Normal subgroup $H$ of $G$ with same orbits of action on $X$.

I have a somewhat broad question related to group actions and their restriction to a normal subgroup. If we have a group action $\sigma : G \times X \rightarrow X$ with orbits $G_x$, and a normal ...
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1answer
22 views

Group actions - modulo 4

I am having a bit trouble understanding group actions. if I am given a set A = {a,b,c,d} and a group action s: Z mod 4 -> $S_A$, how would one then be able to show if there exists a group action s ...
4
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2answers
178 views

group actions on spheres

Let $\mathbb{Z}/2$ act on the $m$-sphere $S^m$ freely and properly discontinuously. If the action is not trivial, can we conclude that the action is homotopy equivalent to the antipodal action? That ...
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1answer
64 views

Transitive actions on sets.

Does there exist a transitive action of $S_4$ on the set $\{1,2,3,4,5\}$ ? I would say no, because the cardinality of our set is bigger than $4$, but I am not sure how to prove this. My suggestion ...
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0answers
46 views

Moment map in general

Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if ...
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2answers
38 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then every element outside of $N$ fixes at most $p$ $N$-orbits.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Also assume that for $g \notin ...
2
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1answer
37 views

The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$

Let $G$ be a permutation group acting transitively on $\Omega$ and suppose $N \unlhd G$ is a normal subgroup of $G$. Assume that for $g \in N_g(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\alpha}^g = ...
2
votes
1answer
26 views

Sign convention when commuting shifts and tensor product

In Markl, Schnider, and Stasheff's Operads in algebra, topology, and physics, they give the observation $\mathfrak{s}^{-1} \mathcal{E}nd_V \cong \mathcal{E}nd_{V[-1]}$ (lemma 3.16) as motivation for ...
1
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1answer
34 views

The kernel of an action on blocks, specifically the action on the orbits of normal subgroup

Let $G$ be a permutation group acting transitively on some set $\Omega$ and suppose we have a normal subgroup $N \unlhd G$. Then the orbits of $N$ form a system of blocks, and if $\Delta$ is such an ...
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0answers
29 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G_{\alpha}N$ is normal if we have $p$ orbits of $N$.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either has no fixed point or exactly $p$ fixed points. Suppose that for $g \notin N_G(G_{\alpha})$ we ...
2
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0answers
37 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ is maximal normal. Then $|G/M| = p$.

Let $G$ be a transitive permutation group on $\Omega$ which fulfills the following property (P) (P) each nontrivial element fixes no point or exactly $p$ points. for some odd prime $p$. Further ...