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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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
29 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
52 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
20 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
34 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
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0answers
31 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
66 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
47 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
55 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
57 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
104 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 ...
1
vote
1answer
25 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
87 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
199 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
413 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
87 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
53 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
38 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
358 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
38 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
82 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
53 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
42 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
91 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
61 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
51 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
83 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
37 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
160 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
49 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
114 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
51 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
43 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
39 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
74 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
97 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
41 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 ...
-1
votes
2answers
124 views

Group acting on a set.

Let $G$ be a group of order $7$ acting on a set of $5$ elements. Show that the action of $G$ must have a fixed point.
0
votes
1answer
53 views

A combinatorial action of a discrete group is proper if and only if it has finite vertex stabilizers

First, let me fix some definitions. The action of a group $G$ on a topological space $X$ is proper if for every compact subspace $K \subseteq X$ the set $\{g \in G \ | \ g K \cap K \neq\varnothing ...
1
vote
1answer
58 views

Question on equivariant functions and subconjugacy

I proved the following proposition as an exercise: Suppose $H \leq K \leq G$ are groups and that $G$ acts on $\frac{G}{H}$ and $\frac{G}{K}$. If $H$ is subconjugate to $K$ (i.e., if $\exists g \in ...
0
votes
1answer
37 views

Determining whether this is a group action

I'm having trouble with an exercise we were given. I have to determine for which values $a,b\in\mathbb{R}$ $$n\cdot t=\phi_n(t)=2^nt+a^n+b$$ defines a group action of the group $(\mathbb{Z},+)$ on ...
1
vote
2answers
122 views

Group Action Questions?

We discussed Group Actions in my undergraduate Modern Algebra class today. I understand the definition and example we went over in lecture, but the problem set is proving difficult. If I want to ...
2
votes
1answer
58 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
0
votes
1answer
64 views

Group action with finite stabilizer.

Let $G$ be a group generated by $\{g_1,g_2,\ldots , g_n\}$. Let $X$ be a space with a $G$-action on it, i.e. $G$ is acting on $X$. Suppose for each $x\in X$, the set $\{g_i;g_i(x)=x\}$ is trivial. ...
5
votes
1answer
141 views

Two subgroups $H_1, H_2$ of a group $G$ are conjugate iff $G/H_1$ and $G/H_2$ are isomorphic

Let $H_1$ and $H_2$ be subgroups of some group $G$. Prove that the left $G$-sets $G/H_1$ and $G/H_2$ are isomorphic (as left $G$-sets) iff the subgroups $H_1$ and $H_2$ are conjugate. If $H_1$ ...