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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4
votes
1answer
71 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) $(...
2
votes
2answers
75 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 $...
1
vote
1answer
75 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:- $...
0
votes
0answers
42 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 $0$, $i$, $−i$, ...
1
vote
0answers
42 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 ...
1
vote
0answers
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 ...
0
votes
1answer
55 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 ...
0
votes
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 $V$. ...
3
votes
0answers
49 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 $$\eta:[X/...
1
vote
3answers
56 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 ...
5
votes
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 ...
2
votes
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 $$r\cdot(...
3
votes
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 ...
1
vote
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 ...
1
vote
2answers
50 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 \...
0
votes
0answers
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 \...
0
votes
0answers
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. ...
1
vote
0answers
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 ...
1
vote
0answers
20 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 ...
0
votes
0answers
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 ...
4
votes
0answers
160 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 ...
1
vote
1answer
24 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
votes
2answers
179 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 ...
1
vote
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 ...
3
votes
0answers
48 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 ...
0
votes
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 N_G(G_{\...
2
votes
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
27 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
vote
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 $...
0
votes
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
votes
0answers
38 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 ...
0
votes
0answers
31 views

What does it mean that the Frobenius representation of a group is unique, and what are its consequences

For a Frobenius group its kernel is a characteristic and nilpotent group, the last property restricts the possibilities how a given group could be represented as a Frobenius group. A statement of this ...
1
vote
1answer
29 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, then maximal normal subgroups are Frobenius groups

Let $p$ be an odd prime and let $G$ be a solvable, transitive permutation group such that each nontrivial element fixes no points or exactly $p$ points on a set $\Omega$. Further suppose that for $g \...
0
votes
0answers
35 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G$ acts on the $N$-orbits in the same way.

Let $G$ be a finite transitive permutation group on $\Omega$ such that every nontrivial element either fixes no point of $\Omega$ or fixes exactly $p$ points of $\Omega$. Assume that for $g \notin ...
0
votes
2answers
39 views

What does it mean to say a single element acts semi-regularly

Let $G$ be a group acting on some set $\Omega$. Just a minor point, but saying that some nontrivial element $g \in G$ acts semi-regularly, does this mean that $g$ itself has no fixed point, or that ...
0
votes
0answers
33 views

If $P\unlhd G$ is semiregular and $G$ such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Each $g \ne 1$ has at most one fixed point in each $P$-orbit

Let $G$ be a finite group acting nonregular and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. Suppose that $N$ is a proper normal $2'$-...
0
votes
1answer
34 views

If $G$ acts nonregular, transitive and $|\mbox{fix}(g)| \le 2$, $|G_{\alpha}|$ is odd and $|\Omega|$ is even, then $|G|$ has twice odd order

Let $G$ act nonregular and transitive on $\Omega$ such that each nontrivial element has at most two fixed points. Let $\alpha, \beta\in \Omega$ be distinct and such that $U := G_{\alpha}\cap G_{\beta}...
1
vote
0answers
53 views

If $G$ acts such that $|\mbox{fix}(g)|\le 2$ for $g \ne 1$ and $O_p(G) \ne 1$ and $|G_{\alpha}|$ odd. Assertions about Frobenius groups

Let $G$ be a finite group acting nonregularly and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. I know three facts: i) If $1 \ne X \le G_{...
0
votes
2answers
14 views

group actions - show $s_2 = e_A$

I'm having a bit of trouble understanding this problem. I am given a set $A = \{a,b,c,d\}$, and a group action $s : \Bbb Z_4 \to S_A$ where the group operation for $\Bbb Z_4$ is addition modulo $4$. I ...
0
votes
0answers
22 views

For a specific subgroup $N$ of index $2$ why $( S \setminus Q ) \cap N = \emptyset$ if $S \in \mbox{Syl}_2(G)$ and $|S : Q| = 2$

Let $G$ be a finite group acting on some set $\Omega$ with $|\Omega|$ even. Let $S \in \mbox{Syl}_2(G)$. Further let $Q \le S$ such that $|S : Q| = 2$ and suppose we have some $x \in S \setminus Q$. ...
1
vote
1answer
25 views

Closed maps and finite group actions

Let $S$ be a topological space equipped with an action of finite group $G$. Let $κ$ be the quotient map. Take some $T⊂S$ such that $κ(T)=κ(S)$ and for each $g\in G$ either $gT=T$ or $gT∩T=∅$. Take ...
1
vote
1answer
57 views

Spivak's curious thoughts about the action of permutations.

Here is an excerpt of Spivak's Differential Geometry. What I do not understand is why he believes $\sigma \cdot (\rho \cdot v) = (\rho\sigma) \cdot v$. Since $\sigma$ and $\rho$ are elements of ...
1
vote
0answers
35 views

For a group acting such that each nontrivial element has at most two fixed point, size of orbit of single $2$-power order element

Let $G$ be a nonregular, transitive permutation group on $\Omega$ such that each nontrivial element has at most two fixed points. Suppose $S \in \mbox{Syl}_2(G)$ and that we have $\alpha, \beta \in \...
0
votes
1answer
62 views

Full flag $Fl_{\mathbb C}(3)$

How we can see that the full complex flag when $n=3$ is equivalent to one of these spaces: $\{(u,v)\in \mathbb CP^2\times \mathbb CP^2 ; u\perp v\}$ and what is dimension over $\mathbb C$ here? $\{(...
0
votes
1answer
19 views

Show that $P$ contains a subgroup of ordre $p^t$ - Use Cauchy's Theorem and the proposition $7.2$

Let $P$ a group of order $p^s$ ($p$ is prime) and $t \leq s$. Use Cauchy's Theorem and the proposition $7.2$ for showing P contains a subgroup of order $p^t$. Cauchy theorem : (1) Let $G$ a ...
1
vote
2answers
25 views

Show that, for each finite $G$-set ($G$-action) on $X$, we have $|X| \equiv |X^G| \pmod n$

Let $G$ a group and $n \in \mathbb{Z}_{>0}$ an integers with the following properties. For each subgroup $H < G$ such that $H \not= G$, the integer $n \mid [G:H]$. Show that, for each finite $...
-2
votes
2answers
44 views

It is group or not? [closed]

Show that the set of vectors defined as directed line segments does not form a group (1) with respect to scalar product (2) with respect to vector product.
1
vote
1answer
35 views

Action of GL(2n,R) on set of linear complex structures

A linear complex structure on $\mathbb R^{2n}$ is an endomorphism $J: \mathbb R^{2n} \to \mathbb R^{2n}$ such that $J^2 = -Id$. (Then $J$ is necessarily an isomorphism.) We have an action of $GL(2n,\...
0
votes
1answer
31 views

Action of permutation group on set of numbers is transitive

I would appreciate if someone could please tell their opinion about my proof. I think the proof makes sense, but I don't know if it's rigorous enough. Theorem: Let $S_n$ be a symmetric group of ...
0
votes
1answer
23 views

low-dim unitary groups and their actions

I need someone to explain for me the unitary groups $U(1)$, $U(2)$ and $U(3)$ and their actions: Specifically: $U(3)/U(2)$ $U(3)/U(2)\times U(1)$ $U(3)/U(1)\times U(1) \times U(1)$ I have seen ...