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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2
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0answers
40 views

Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
1
vote
1answer
45 views

Orbits of action of $SL_2(\mathbb{Z})$ on lattice

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$. Specifically, what are the orbits of this action?
1
vote
0answers
20 views

Circle action on the product of a Mobius band and a circle.

Consider the product of a Möbius band and a circle $Mo\times S^1$. Is there a circle action on $Mo\times S^1$ such that it is equivariantly homeomorphic to the twisted product $D^2 ...
5
votes
1answer
77 views

If a finite group $|G|$ acts transitively on a set $X$ with $|X|=2^n$, $n \geq 1$, then $G$ has an involution with no fixed points

Let $G$ be a finite group acting transitively on a set $X$, where $|X| = 2^n$ for some $n \geq 1$. Show that some element of $G$ acts as an involution with no fixed points. While it is fairly easy ...
0
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0answers
210 views

Can we prove $H \cong xHx^{-1}$ given $H \le G, x \in G$ using group action?

The exercise is as follows: $G$ is a group, $H \le G$. For any $x \in G$, to prove that $H \cong xHx^{-1}$. I am able to prove this isomorphism by defining a bijection $f : h \mapsto xhx^{-1}$ ...
0
votes
0answers
7 views

$\bar{L}$ points of $GL_{(ab)^2}/PGL_a\times PGL_b$

I am reading the paper "Matrix invariants of composite size" by A. Schofield (see here), and I have trouble understanding some one of his arguments, and I hope someone can explain them to me. He ...
0
votes
0answers
23 views

How to write Cayley representation permutations as cycles?

The representation of $g \rightarrow xg$ was given to us as Cayley representation. I believe it means that every group element $g \in G$ is mapped to another element $xg$ where $x$ is the same for ...
0
votes
1answer
23 views

Compute $S_3$ acting by conjugation on the set $X$ of $6$ subgroups of $S_3$

I know that the subgroups of $S_3$ are $\{e\}$, $\langle(12)\rangle$, $\langle(13)\rangle$, $\langle(23)\rangle$, $A_3$, and $S_3$. What I also know is that conjugation is $C_g(H) = gHg^{-1}$. Thus in ...
2
votes
0answers
18 views

Finding a fundamental polygon for two-generator subgroup of PSL(2,R)

Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like ...
0
votes
0answers
31 views

Action of a Lie group, a map of constant rank

Consider some Lie group $G$, smooth manifold $X$ and some action of $G$, i.e. a group homomorphism $\mathcal{A}: G\longrightarrow \mathrm{Diffeo}(X)$ such that the map $(g,x)\mapsto ...
0
votes
0answers
17 views

six transitive permutation groups

If I'm explaining right than please give me some hints about how we prove a permutation group is six transitive. I have proved that it is two transitive because stabilizer of one point acts ...
0
votes
1answer
33 views

Question about May's Algebraic Topology book

I am referring Google Books for the question: link in the proof of the first lemma, why is $hns=\phi(hs)$ true? I simply cannot get it...
3
votes
0answers
29 views

Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
1
vote
0answers
45 views

Notation for pointwise versus “setwise” stabilizers

Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ...
1
vote
1answer
45 views

Is the stabilizer of an element $\delta$ in the stabilizer of $\omega$ in G equal to the pointwise stabilizer of $\{ \delta, \omega \}$

i.e., is $(G_{\delta})_{\omega} = G_{( \{\delta, \omega\} )}$? I know that \begin{eqnarray*} (G_{\delta})_{\omega} &=& \{ \forall g \in G_{\delta} \,|\, \omega^g = \omega \} \\ &=& ...
2
votes
1answer
40 views

How to prove that $N$ is 2-transitive on $\Omega$?

Suppose $\Omega$ is a finite set with $|\Omega| \geq 5$. Let $G$ act faithfully on $\Omega$ such that $G$ is 4-transitive on $\Omega$. Let $N$ be a normal, nontrivial, nonregular subgroup of $G$. I ...
0
votes
2answers
47 views

What is the centralizer of (1 2 3)(4 5 6) in $S_6$

So far, I've seen that the following permutations are in the centralizer: $(1 4), (2 5), (3 6)$, products of these transpositions(EDIT: not all of these are in the centralzier), $(1 2 3), (1 3 2), (4 ...
2
votes
1answer
90 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
0
votes
0answers
52 views

What is an algebraic automorphism over $k$?

I'm reading some notes about the action of finite groups on algebraic varieties, and I've found this sentence. Let $Y$ be a scheme of finite type over a field k, and let $G$ be a finite group, ...
1
vote
1answer
24 views

a question concerning subgroup of symmetric group

Suppose $H$ is a transitive subgroup of the symmetric group of $n$ symbols. Show that $n$ divides the order of $H$. I tried to show that some $n$-cycle is in $H$ but this idea did not work.
0
votes
2answers
73 views

Show that group action is homomorphism to Symmetric group

I'm just barely getting my feet wet with abstract algebra, currently working on understanding group action. According to the wikipedia article, a group action $A$ of group $G$ on set $X$ is a group ...
0
votes
2answers
26 views

the cardinal of $x^G$ factors the cardinal of $x^N$

please give me hints to solve this problem: Let $G$ acts on $X$ and $N$ be a normal subgroup of $G$, show that for every $x\in X$ we have: the cardinality of $x^G$ factors the cardinality of $x^N$ .
3
votes
2answers
157 views

Groups acting on schemes: the quotient scheme doesn't always exist.

Preliminary notion: Consider the action of a group $G$ on an object $X$ of some category $\mathcal C$. We have a group homomorphism $\rho:G\longrightarrow\operatorname{Aut}(X)$ which sends $g$ in ...
10
votes
3answers
407 views

Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
2
votes
2answers
81 views

Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
0
votes
1answer
45 views

Transitive group action restricted to normal subgroup

Let $G$ be a finite group, and let $\Omega$ be a transitive $G$-space. Assume 1 $\neq H \unlhd G$ and that |$\Omega$| = $p$ where $p$ is prime, and $G \leq Sym(\Omega)$. Deduce that then $H$ must act ...
0
votes
2answers
37 views

Embedding monomorphism between Symmetric Groups

Suppose that $m$ and $n$ are positive integers, and $m<n$. Define $I:S_m \rightarrow S_n$ as follows: Given $\alpha \in S_m$, we let $\hspace{150pt}I(\alpha)(k)=\alpha(k) ...
5
votes
5answers
306 views

Poincaré's theorem about groups

Let $G$ be a group and $H<G$ such that $[G:H]<\infty$. There exists a subgroup $N\triangleleft G$ such that $[G:N]<\infty$. I have to show this fact (that according to my book is due to ...
1
vote
1answer
32 views

Inequality regarding orbits of groups

I've been working on a question for a few days now, and I'm stuck on proving a claim that I don't know if there's any reason for it to be true. I'll write it here in the greatest generality I can ...
1
vote
0answers
37 views

How we show primitive action shows alternating group

I have a graph (as shown in figure), which represents a quotient of the group $$G=\langle A,B,C,D; A^3=B^2=C^3=D^2=(AC)^2=(AD)^2=(BC)^2=(BD)^2=1 \rangle.$$ I proved that $G$ acts 2-transitively and so ...
3
votes
1answer
81 views

Is there an easy way to tell if these two SO(2)s in SO(4) are conjugate?

I am currently interested in quotients of Lie groups by submaximal tori. $G = Sp(1) \times Sp(1)$ double-covers $SO(4)$, as noted at The Quaternions and $SO(4)$. Define a circle subgroup $T = \{1\} ...
3
votes
0answers
52 views

Dimension of a constructible set intersecting each orbit of a $G$-variety

In preparing a talk I'm having trouble with exercise 3 and 4 on page 25 of the following Lecture Notes of Crawley-Boevey (I only need the case $X=Y$ there): $\text{3.}$ Let $X$ be a variety ...
0
votes
0answers
38 views

“Fundamental region” for non-discrete Moebius groups.

Suppose we are given a discrete, faithful representation $\rho$ of $F_2=\langle a,b|\rangle$, the free group on two generators, into $\mathbb{P}SL(2,\mathbb{R})$, so that the quotient is homeomorphic ...
2
votes
1answer
55 views

Orbits that 'coalesce'

Let $R$ be a commutative ring, $G$ a group scheme over $\mathrm{Spec}\;R$, and $X$ a scheme over $\mathrm{Spec}\;R$ on which $G$ acts $R$-morphically via $G\times X\to X$. Suppose $S$ is another ...
1
vote
1answer
84 views

What are the conjugacy classes in $\mathrm{Aut}(G)$?

Let $G$ be an arbitrary group, and let $\mathrm{Aut}(G)$ be the group of automorphisms of $G$ (with composition of morphisms as multiplication). I'd like to learn more about the problem of ...
0
votes
1answer
51 views

The anti-symmetrization and simetrization operators are mutually orthogonal

For each vector $x=(x_1,\dots,x_n)$ of an $n$-dimensional vector space $V$, and for each permutation $s$ of the symmetric group on the $n$-element set $S_n$, put $s(x)=(x_{s(1)},\dots,x_{s(n)})$. Then ...
1
vote
1answer
44 views

What motivates the definition of “Periodic” group action

Consider a group $G$ acting on a set $\Omega$. For example, let $G=\{g\in A(\mathbb R):(\alpha +1)g=\alpha g+1\}$ for all $\alpha\in\mathbb R$, where $A(\mathbb R)$ are the order-preserving ...
1
vote
1answer
79 views

Group action on set of maps - formula

It is given that $G:X$ and $G:Y$. Does this $[g\bullet f](x) := g\bullet f(g\bullet x)$ formula define group action $G:(Y^{X})$ I guess it doesn't, but I can't prove it as for now. And there must be ...
0
votes
1answer
36 views

Notation for permutation corresponding to the action of a group element

Let $G \times X \to X,\ \ (g,x) \mapsto g.x$ be an action of $G$ on $X$, i.e., $e.x = x$ for all $x \in X$; $gh.x = g.(h.x)$ for all $g \in G$, $x \in X$. For a fixed $g \in G$, how should I refer ...
8
votes
0answers
157 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
0
votes
0answers
30 views

Finding conjugacy classes of $D_{10}$

Looking at the group $D_{10}$, I have found that for some (non-identity) rotation $\rho$ its centraliser has order 5, and for some reflection $\tau$ its centraliser has order 2. By the ...
3
votes
0answers
45 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
0
votes
4answers
108 views

Prove that the number of elements of every conjugacy class of a finite group G divides the order of G.

Prove that the number of elements of every conjugacy class of a finite group $G$ divides the order of $G$. I'm studying for my Group Theory exam and this was a question on a previous exam. I ...
1
vote
1answer
46 views

Express $ G_y$ in terms of $G_x$. [duplicate]

A finite group $G$ acts on a finite set $X$, the action of $g \in G$ on $x \in X$ being denoted by $gx$. For each $x \in X$ the stabilizer of $x$ is the subgroup $G_x = \{g \in G : gx = x\}$. If $x, y ∈ ...
1
vote
1answer
40 views

Group theory: group actions on finite group.

I'm having trouble with the following question: Let $G$ be a finite group acting on a finite set $X$. For $g\in G$, let $Fix_X(g) =\{x\in X \mid xg = x\}$ and, for $x\in X$, let $G_x = \{g\in G \mid ...
1
vote
2answers
69 views

Counterexample that $a\in G$, $a^n\notin H$, for $H$ a subgroup of finite index $n$ in $G$. [duplicate]

Let $G$ be a group and $H$ a subgroup of finite index $n$. Give a counterexample that $a\in G$, $a^n\notin H$ (although I can prove that there exists $k\in\{1,2,\dots,n\}$ such that $a^k\in H$). ...
4
votes
1answer
37 views

Graph with sharply 1-transitive automorphism group

What finite Graphs $G$ have the property that for all $v,w\in G$, there is exactly one automorphism $\phi$ of $G$ with $\phi(v)=w$? Of course, each of the three graphs with one or two vertices have ...
1
vote
1answer
73 views

how should i describe this combination of group actions?

let $A$ be a multiplicative abelian group and let $D_m=D_m(A)$ for integer $m \gt 0$ be the group of $m \times m$ diagonal matrices with entries in $A$. now $D_m$ has a subgroup $A^* \cong A$ which is ...
1
vote
1answer
80 views

If $X$ is $G$-paradoxical then $G$ is $G$-paradoxical. Is my proof correct?

I am currently reading Stan Wagon's Banach-Tarski Paradox book, and this was left as an exercise to prove (converse of Proposition 1.10). Let $X$ be a set, and let $G$ act on $X$ with no ...
1
vote
0answers
35 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...