-1
votes
0answers
24 views

Prove that G is a group acting on a set X. Where G= {(1),(123),(132),(45),(123)(45),(132)(45)} and X= {1,2,3,4,5}

I understand that the axioms that must be satisfied to prove that this is an "action" is: ex = x for all x an element of X (compatibility with identity). g_1(g_2*x) = (g_1g_2)*x (compatibility with ...
1
vote
1answer
32 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
76 views

If G acts on X, show that there must be a fixed point for this action. Please help. [on hold]

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
31 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
70 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
28 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
51 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
36 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 ...
0
votes
0answers
33 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
41 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
39 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
17 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
28 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
24 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
37 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
35 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
42 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
76 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
19 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
51 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
25 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
31 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
34 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
272 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
1answer
79 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 ...
1
vote
1answer
36 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
64 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 ...
0
votes
4answers
86 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
44 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
39 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
68 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
36 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
72 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
75 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
votes
2answers
116 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
47 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 ...
0
votes
1answer
37 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 ...
1
vote
2answers
99 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 ...
0
votes
1answer
47 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
109 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 $H_1$ ...
2
votes
1answer
61 views

Action on Pairs, On Sets and on points in GAP

I am trying to understand GAP in group action. I am confused in few things what is the difference between action on pairs, on sets, with the domain sometimes on list, and on blocks. Please help me to ...
0
votes
2answers
269 views

About stabilizer in group action

Let $X$ be a finite set and $x$ is an element of $X$. Let $G_x$, the stabilizer subgroup, be the subset of $S_X$ consisting of permutations that fix $x$. The question is Is stabilizer always a ...
2
votes
1answer
72 views

When is the action of $G$ on $\text{Syl}_p(G)$ by conjugation is double transitive?

We know that the action of $G$ on $\text{Syl}_p(G)$ by conjugation is transitive. I wonder when this action can be double transitive on $\text{Syl}_p(G)$. Thanks for your help.
5
votes
3answers
283 views

Without using Sylow: Group of order 28 has a normal subgroup of order 7

Prove that a group of order 28 has a normal subgroup of order 7. How can I prove this without using Sylow's theorem? I know by Cauchy’s theorem, there exists an $x\in G$ with order 7, now I just ...
1
vote
3answers
74 views

Relationship between group actions and homomorphisms

I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ...
1
vote
0answers
37 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
0
votes
0answers
61 views

Terminology on group actions

Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. My knowledge of group theory is undergraduate-level stuff. I'm looking at the paper cited ...
7
votes
1answer
100 views

Understanding what an action is?

This is a very simple question, and I am quite embarrassed to ask it! I'm trying to understand what an action is in general, and perhaps the best place to start is to try and outline my current ...