0
votes
0answers
20 views

Prove a result on transitive group actions.

Let $G$ be a group and $A$ & $B$ be two sets s.t. $G$ acts transitively on each of $A$ & $B$. Choose some $\alpha$ and $\beta$ in $A$ & $B$ respectively then prove that if $G=G_\alpha ...
2
votes
2answers
69 views

For a transitive permutation group $G,$ show that there is some $\sigma \in G$ such that $\sigma(a) \neq a$ for all $a \in A.$

Here is a problem that I have been working on. I was able to prove part A, but am having problems with part B. Thanks! Let $G$ be a permutation group acting on a finite set $A.$ If $g\in G,$ let ...
1
vote
2answers
48 views

How to describe $G/U$?

Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, ...
3
votes
4answers
50 views

Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
0
votes
0answers
33 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
1
vote
1answer
43 views

Double cosets and conjugation

Let G be a group and $h,g \in G$ with $SgT=ShT$ Show that the subgroups $gTg^{-1}\cap S$ and $hTh^{-1}\cap S$ are conjugated in S. Two subgroups $U,V$ are conjugated if $\phi(U)=gUg^{-1}=V$, right? ...
0
votes
1answer
38 views

Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
3
votes
2answers
40 views

Index and normal subgroups

I want to show the following. For an infinite group G with only two normal subgroups (G and {e}) holds: There does not exist a non-trivial subgroup of G with finite index. I think i should prove ...
3
votes
1answer
40 views

The index of the Core of a group

I have to prove the following: Let $G$ be a group and $U$ be a subgroup of $G$. Then it holds: If $U$ has finite index, then $\text{Core}_G(U):=\bigcap\limits_{g\in G}gUg^{-1}$ has also finite index. ...
3
votes
0answers
60 views

Orbits of the group action $g. (x,y) = (gx,gy)$ on cartesian product

Let $G$ be a group acting on a set $X$. Then we have a natural action on $X \times X$ in the following way: $g. (x,y) = (gx,gy)$. Then, suppose we have two points of interest $x_1,x_2 \in X$, and we ...
5
votes
1answer
61 views

Recovering a group action from sizes of orbits of individual elements

Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...
3
votes
1answer
51 views

The group acts on an ordered set

Let $G$ be a group, $G$ acts on an ordered set and preserves its order, i.e. $a<b$, then $g(a)<g(b)$ for $g\in G$. Then does it imply there is a left order on $G$, i.e. $f<g$, then $fh<gh$ ...
0
votes
1answer
32 views

the group acts faithfully on the line

Let $G$ be a group. $G$ acts faithfully on the line $\mathbb{R}$ by orientation preserving homeomorphism, then does it imply $G$ is left ordered, i.e. there is an order $<$ on $G$, and if $a<b$, ...
0
votes
1answer
47 views

The action of free group on line

Let $G$ be a free group, if the action of $G$ on $\mathbb{R}$ is free, does it imply that $G$ is abelian?
3
votes
3answers
169 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
1
vote
2answers
34 views

Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a ...
1
vote
1answer
32 views

The function $A:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ given by $(n,x)\to nx$ is a group action on $\mathbb{R}$.

I somehow came through this True or False question, but before answering it, I just wanted to ensure some things. I understand that the axioms for $group-action$ must be met. Basically: ...
2
votes
1answer
33 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
0
votes
0answers
30 views

Standard action of SU(3) on $\mathbb{C}^3$

It is my understanding that SU(3) acts on $\mathbb{C}^3$ the same way SO(3) acts on $\mathbb{R}^3$, i.e. as proper rotations. If this is the case, then the orbit of $(1,0,0) \in \mathbb{C}^3$ ...
1
vote
0answers
62 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
5
votes
4answers
92 views

On what sets can $\mathfrak{S}_n$ act transitively?

I would like to know $\mathfrak{S}_n$ could act faithfully transitively on sets with $m$ elements, with $m > n$. I know that it is not possible if $m = n+1$ except for $n = 5$. Any ideas ?
0
votes
1answer
84 views

How can I use Clebsch-Gordan coefficients to decompose this group representation?

Let $G$ be a compact group, $\alpha$ be a unitary irrep of $G$ with carrier space $\mathcal A$, and $\beta$ be a unitary irrep of $G$ with carrier space $\mathcal B$. Then, the action of $G$ on ...
1
vote
1answer
26 views

Intersection of stabilisers

I have a short question: if $G$ acts on $X$, is the intersection of all the stabilisers the same as the conjugates of the stabilisers? In other words does, for any $z\in X:$ $$\bigcap_{g\in G} ...
2
votes
1answer
20 views

transitively action of stabilizer of G

if $G$ acts transitively on $X$ and for a special $x$ in set $X$ we have the stabilizer of $x$ acts transitively on $X-\{x\}$ can we conclude that this proposition is true for all element of $X$?
1
vote
2answers
39 views

Prove this wreath product is a group [Homework]

I'm not usually one to post unworked problems here... I usually try to at least have an attempt, but unfortunately in this case I'm unable to even get an intuitive sense of what's going on here - and ...
3
votes
1answer
85 views

Why it is a group action?

Let a group $G$ acts on a vector space $V$ and let $f$ be a function on $V$. The action of an element $g \in G$ defined by the rule $g f(x)=f(g^{-1} x), \forall x \in V.$ A typical proof from a ...
2
votes
1answer
38 views

How many orbits are there in the group action $A\colon 3\mathbb{Z} \to\mathbb{Z}_{6}$ with action given by $(3n,m)=(3n+m)\bmod6$.

I am having difficulty trying to understand this question. All I know is that $3\mathbb{Z}$ and $\mathbb{Z}_6$ are both groups, that is: $$3\mathbb{Z}=\{\dotsc, -6, -3, 0, 3, 6 \dotsc\}$$ ...
1
vote
1answer
37 views

show that there is some element x∈X whose stabilizer Gx is all of G where G is a group of order p^k, where p is prime and k is a positive integer

I'm having trouble with this problem: Suppose that G is a group of order p^k, where p is prime and k is a positive integer.
0
votes
3answers
87 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 size |X| of X. If G acts on X, show that ...
0
votes
0answers
44 views

Burnside's Lemma and Stirling Numbers of the First Kind

I've seen that $n!=\displaystyle\sum_{p=0}^n s(n, p)n^p$, where $s(n, p)$ are the signed Stirling Numbers of the First Kind, whose absolute values count the number of permutations in $S_n$ which have ...
2
votes
1answer
81 views

Representation of $GL_2$ on $K^2$

In one of my problems it says the following: Let $K$ be an infinite field. Consider the linear action of $GL_2$ on $K[x,y]$ induced by the natural representation of $GL_2$ on $K^2$. I don't know what ...
1
vote
1answer
31 views

Why are isotropy groups named as such?

Why are isotropy groups, also known as stabilizers, named as such? In physics, the word isotropy means having the same property in all directions. Can one draw an analogy from this to interpret the ...
7
votes
1answer
58 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
2
votes
0answers
41 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?
5
votes
1answer
88 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
votes
0answers
268 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
24 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
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...
1
vote
0answers
49 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
46 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
48 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
91 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
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 $x^N$ .
0
votes
1answer
47 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
323 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 ...