Tagged Questions
2
votes
1answer
26 views
Infinite imprimitive non abelian group?
My new question is
Is there an infinite, imprimitive and non abelian group?
Thank you for the further answers.
2
votes
1answer
42 views
Function spaces and transitive group actions
Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first.
Let $B$ be a topological space and $G$ a topological group ...
0
votes
1answer
55 views
Let $G$ be transitive.Then $\beta\in \operatorname{fix}(G_\alpha)$ implies $G_\alpha = G_\beta$
i am new in this forum.
My question is about group actions
We have a transitive action of $G$ and $\beta$ a element in the fixed points of the stabilizer of another element $\alpha$. Then $\alpha$ ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
10
votes
0answers
72 views
Show that $h \equiv 1 \pmod p$, where $h$ is the number of subgroups of order $p$ and $p$ divides the group order. [duplicate]
Let $G$ be a finite group and $p$ a prime number that divides the
order of $G$. Let $h$ be the number of subgroups of $G$ of order $p$.
Prove that there are $h(p-1)$ elements of order $p$ in ...
2
votes
0answers
37 views
Semi-orbital equivalence relation
Edit: I was in kind of a hurry when writing this post and made a mistake in the formula defining $G_E$. What I had written said that $G_E$ preserves the set of classes of $E$, while I meant actually ...
4
votes
2answers
70 views
Free objects in $\mathrm{Set}(G).$
What are the free objects in the category of $G$-sets for a group $G$?
After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
4
votes
2answers
69 views
About the category $\mathrm{Set}(G)$
I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
0
votes
1answer
25 views
Need to show that order of orbits under group action is non-trivial and intersection of two p-groups is a proper subgroup
I'm working my way through the second and third sylow theorems in my book. Here's the relevant bit:
We have a group $G$ of order $p^\alpha m$ where $p$ does not divide $m$. We have that $Q$ is a ...
1
vote
1answer
26 views
Question about Sylow's Theorem/Conjugation of the set of conjugates of P
I'm trying to understand a proof of the 2nd and 3rd parts of Sylow's Theorem. In some preliminary work, my book establishes that $P$ is a Sylow p-subgroup of $G$.
Then it defines ...
1
vote
1answer
32 views
Studying the action of $GL(V)$ on the vector space $V$
The statement I am trying to prove is the following.
Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let
$\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural
...
3
votes
1answer
25 views
Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types.
Show that the orbits of $S_n$ under the conjugation action of $S_n$
on itself correspond 1-1 with the cycle types.
So, the orbit of $\sigma \in S_n$ is the set $S_n \sigma = \{ \tau .\sigma : ...
3
votes
1answer
55 views
Conjugation on subgroups of $A_4$ faithful?
Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$
I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
4
votes
0answers
42 views
Equiv Relation of Orbits - Group Action [duplicate]
Let $G$ be a group that acts on $X$. I want to show that the orbits of $G$ partition $X$. I am given the relation $x\sim y \iff x\in Orb(y)$. Now:
$x\sim y\iff x\in Orb(y) \iff x=gy$ for some $g\in G ...
1
vote
1answer
42 views
right group action
wikipedia says 'The difference between left and right actions is in the order in which a product like $gh$ acts on $x$. For a left action $h$ acts first and is followed by $g$, while for a right ...
4
votes
3answers
53 views
at least one element fixed by all the group
$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$?
I tried to ask this question ...
3
votes
0answers
36 views
burnside lemma cube [duplicate]
Having n colors, use the lemma to find a formula for the number of ways to color the edges of the cube.
What I have so far:
I got $|A/G| = \dfrac{n^{12} + 6n^3 + 3n^6 + 8n^4 + 6n^7}{24}$ but when I ...
2
votes
1answer
108 views
Using Burnside's lemma on the cube.
Having $n$ colors, use the lemma to find a formula for the number of ways to color the edges of the cube.
Here is what I have so far: The Burnside lemma says that $\displaystyle |X/G| = ...
0
votes
1answer
80 views
p-group and group actions
$G$ is a $p$-group, which means $|G|=p^n$ for $n\in \mathbb{Z^+}$.
Now,if $p$ does not divide $|S|$, for S is a set that G acts upon, how do I show that there exists $a\in S$ such that $G_a=G$
So ...
2
votes
1answer
56 views
What does “lifted action” mean?
I read about angular moment and linear moment but I don't know what "lifted action" means. Can you explain please?
Thanks. :)
0
votes
2answers
53 views
Group and orbit question.
Suppose group $G$ acts on a set $A$.
a) If $x$ and $y$ are in the same orbit, show that there exists some $g \in G$ such that $gG_x g^{-1} = G_y$.
b) Show that if $|G.x|$ is finite, then $|G.x| = ...
6
votes
1answer
102 views
Group actions transitive on certain subsets
Let $G$ be a group acting on a finite set $X$. This also gives an action of $G$ on the subsets of $X$ of any given size, and we can ask whether this action is transitive for some specified size of ...
-3
votes
2answers
126 views
On Conjugacy Classes and Alternating Group $A_n$
in Dummit & Foote
in page 131
"
Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ .
1.Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any odd ...
4
votes
3answers
82 views
Do we gain anything interesting if the stabilizer subgroup of a point is normal?
Let $G$ be a group and $S$ a $G$-set with action $(g,s) \mapsto gs$. For some $s \in S$, let the stabilizer of $s$, $G_s=\{g \in G\,|\,gs=s\}$ be normal in $G$. What does this let us say about the ...
1
vote
2answers
169 views
Rotational Symmetries of a Cube
Use the Orbit Stabilizer Theorem to deduce the number of elements in the rotational symmetry group of the cube.
I can write $\operatorname{Stab}_G(v) = \left\{g \in G \mid g \cdot v = v\right\}$ and ...
3
votes
3answers
110 views
Examples of the dihedral group $D_4$ acting on sets
Consider the group $D_4$. Give examples of $D_4$ acting on a set.
Attempt: So $|D_4| = 8$. I have come up with a few, but I was wondering what some people here thought.
First one we came up with ...
0
votes
0answers
60 views
Actions of G on corresponding orbits are equivalent for stable maps
Given actions of G on X and on Y, these actions are equivalent if and only if there is a bijection from $X $ \ $ G \rightarrow Y$ \ $G$ so that actions of G on corresponding orbits are equivalent.
...
1
vote
2answers
66 views
Finding the number of orbits
How many orbits are there of $(12)(25)$ in $S_{5}$?
Considering the permutation $(12)$, it has $4$ orbits and is as follows:
$\{\{1,2\},\{3\},\{4\},\{5\}\}$
and (25) also has 4 orbits and is also ...
3
votes
1answer
62 views
The order of a conjugacy class is bounded by the index of the center
If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)
Here is my attempt: we have
...
1
vote
2answers
54 views
$|G|=11$ operates on $\mathbb{Z} / 5\mathbb{Z}\times \mathbb{Z} / 5\mathbb{Z}$ implies at least one point fixpoint
Can someone please help me with the following: A group of order $11$ operates on $\mathbb{Z} / 5\mathbb{Z}\times \mathbb{Z} / 5\mathbb{Z}$. I have to show that it has at least one fix point. Can ...
1
vote
2answers
92 views
Properties of set $\mathrm {orb} (x)$
Properties of set $\mathrm {orb} (x)$:
${\displaystyle \bigcup_{x\in X}\mathrm{orb}(x)=X}$;
$\mathrm{orb}(x)\cap\mathrm{orb}(y)=\emptyset$
for all $x,y\in X, x\neq y$
How to prove it? Please ...
1
vote
1answer
34 views
According to my solution there should be more fixed points…
I have solved the following exercise:
A group of order $55$ acts on a set of order $18$. Then there are at least $2$ fixed points.
But according to my solution, there should be at least 3 fixed ...
3
votes
1answer
114 views
On Group action and blocks of subgroups of the symmetric Group
this exercise is from Dummit and foote , page 117 , # 7.d
prove : a transitive group $G$ is primitive on $A$ iff for each $a \in A$ , the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$
...
2
votes
1answer
82 views
In a group of Möbius transformations, does discontinuity imply discreteness?
Let $G$ be a subgroup of the group of Möbius transformations
$$ z \mapsto \frac{az+b}{cz+d}.$$
What is the relationship between the two conditions:
(1) $G$ being discrete.
(2) $G$ acting properly ...
2
votes
3answers
150 views
Exercises about group actions
I am having trouble with exercises from Chapter IV of Aluffi's Algebra: Chapter 0. After spending many hours on the following two, I guess I need some help:
Let $G$ be a finite group, and suppose ...
1
vote
1answer
71 views
determine stabilizer of an edge of the cube and its orbit
Let $G$ be the group of symmetries of the cube, and consider the action of $G$ on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of $G$.
...
2
votes
2answers
324 views
Orbit, stabilizer and fixed points of a group's action on left cosets by left multiplication
For this example - with hardly any steps - I'm not getting its conclusions with my work. Can someone please see where I might have gone wrong? I added the colors. Thank you.
Given Example
$G$ ...
0
votes
3answers
142 views
Faithfulness - Group Action on Left Cosets by Left Multiplication
A lecture stated that if $ H \leq \text{ group } G $ and $ G $ acts on $ \{gH\} = G/H $ by left multiplication, then this left multiplication action of $ G \text{ on } {gH} = G/H $ is faithful iff $ ...
5
votes
3answers
108 views
Group Action - Permutation on the Polynomial
I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...
6
votes
1answer
84 views
Invariants of binary forms under a $\begin{pmatrix} 1& 1 \\ 0& 1 \end{pmatrix}$ action
The special linear group $\text{SL}_2(\mathbb{Z})$ of $2\times 2$ invertible matrices in $\mathbb{Z}$ acts on binary cubic forms $\{ax^3 + bx^2y + cxy^2 + dy^3\}$ by acting on the vector $(x,y)^T$. ...
2
votes
1answer
68 views
Motivation for the term “transitive” group action
I have two questions:
In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not ...
1
vote
1answer
41 views
Definition of tautological action
What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions?
For reference the particular sentence is: "A ...
3
votes
2answers
99 views
Original papers on the subject of group actions
Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. ...
0
votes
2answers
66 views
Composite group homomorphism between alternating groups
Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
3
votes
2answers
149 views
Transitive action of normal subgroup of the alternating group
everyone! Would anyone be willing to give me any sort of help with the following question?
Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the ...
4
votes
0answers
147 views
Semi-direct product isomorphic to direct product
I would like some help on the following problem from anyone who would like to help.
Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$.
The situation ...
0
votes
3answers
72 views
A step in the proof of Cauchy's theorem for groups
Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
0
votes
1answer
93 views
Injective Homomorphism on $S_n$
I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody ...
1
vote
1answer
62 views
Showing equality of two groups.
Consider $S_n$, the permutation group. Let $a\in S_n$. I want to show that if $a$ is an $n$ or an $n-1$ cycle, then $\langle a\rangle = \{c \in S_n : ca=ac\}$. Any help will be veru much appreciated.
1
vote
0answers
57 views
Isomorphism between G-equivariant bijections and Normalizer
If anyne could give me some help with this, it will be deeply appreciated:
Let $X$ be a set equipped with the action of some group $G$. Denote by $Aut_G(X)$ the set of $G-equivariant$ bijections $f: ...
