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.

learn more… | top users | synonyms

0
votes
0answers
30 views

What does it mean that the Frobenius representation of a group is unique, and what are its consequences

For a Frobenius group its kernel is a characteristic and nilpotent group, the last property restricts the possibilities how a given group could be represented as a Frobenius group. A statement of this ...
1
vote
1answer
27 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, then maximal normal subgroups are Frobenius groups

Let $p$ be an odd prime and let $G$ be a solvable, transitive permutation group such that each nontrivial element fixes no points or exactly $p$ points on a set $\Omega$. Further suppose that for $g ...
0
votes
0answers
35 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G$ acts on the $N$-orbits in the same way.

Let $G$ be a finite transitive permutation group on $\Omega$ such that every nontrivial element either fixes no point of $\Omega$ or fixes exactly $p$ points of $\Omega$. Assume that for $g \notin ...
0
votes
2answers
37 views

What does it mean to say a single element acts semi-regularly

Let $G$ be a group acting on some set $\Omega$. Just a minor point, but saying that some nontrivial element $g \in G$ acts semi-regularly, does this mean that $g$ itself has no fixed point, or that ...
0
votes
0answers
33 views

If $P\unlhd G$ is semiregular and $G$ such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Each $g \ne 1$ has at most one fixed point in each $P$-orbit

Let $G$ be a finite group acting nonregular and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. Suppose that $N$ is a proper normal ...
0
votes
1answer
34 views

If $G$ acts nonregular, transitive and $|\mbox{fix}(g)| \le 2$, $|G_{\alpha}|$ is odd and $|\Omega|$ is even, then $|G|$ has twice odd order

Let $G$ act nonregular and transitive on $\Omega$ such that each nontrivial element has at most two fixed points. Let $\alpha, \beta\in \Omega$ be distinct and such that $U := G_{\alpha}\cap ...
1
vote
0answers
53 views

If $G$ acts such that $|\mbox{fix}(g)|\le 2$ for $g \ne 1$ and $O_p(G) \ne 1$ and $|G_{\alpha}|$ odd. Assertions about Frobenius groups

Let $G$ be a finite group acting nonregularly and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. I know three facts: i) If $1 \ne X \le ...
0
votes
2answers
14 views

group actions - show $s_2 = e_A$

I'm having a bit of trouble understanding this problem. I am given a set $A = \{a,b,c,d\}$, and a group action $s : \Bbb Z_4 \to S_A$ where the group operation for $\Bbb Z_4$ is addition modulo $4$. I ...
0
votes
0answers
22 views

For a specific subgroup $N$ of index $2$ why $( S \setminus Q ) \cap N = \emptyset$ if $S \in \mbox{Syl}_2(G)$ and $|S : Q| = 2$

Let $G$ be a finite group acting on some set $\Omega$ with $|\Omega|$ even. Let $S \in \mbox{Syl}_2(G)$. Further let $Q \le S$ such that $|S : Q| = 2$ and suppose we have some $x \in S \setminus Q$. ...
1
vote
1answer
25 views

Closed maps and finite group actions

Let $S$ be a topological space equipped with an action of finite group $G$. Let $κ$ be the quotient map. Take some $T⊂S$ such that $κ(T)=κ(S)$ and for each $g\in G$ either $gT=T$ or $gT∩T=∅$. Take ...
1
vote
1answer
56 views

Spivak's curious thoughts about the action of permutations.

Here is an excerpt of Spivak's Differential Geometry. What I do not understand is why he believes $\sigma \cdot (\rho \cdot v) = (\rho\sigma) \cdot v$. Since $\sigma$ and $\rho$ are elements of ...
1
vote
0answers
35 views

For a group acting such that each nontrivial element has at most two fixed point, size of orbit of single $2$-power order element

Let $G$ be a nonregular, transitive permutation group on $\Omega$ such that each nontrivial element has at most two fixed points. Suppose $S \in \mbox{Syl}_2(G)$ and that we have $\alpha, \beta \in ...
0
votes
1answer
60 views

Full flag $Fl_{\mathbb C}(3)$

How we can see that the full complex flag when $n=3$ is equivalent to one of these spaces: $\{(u,v)\in \mathbb CP^2\times \mathbb CP^2 ; u\perp v\}$ and what is dimension over $\mathbb C$ here? ...
0
votes
1answer
19 views

Show that $P$ contains a subgroup of ordre $p^t$ - Use Cauchy's Theorem and the proposition $7.2$

Let $P$ a group of order $p^s$ ($p$ is prime) and $t \leq s$. Use Cauchy's Theorem and the proposition $7.2$ for showing P contains a subgroup of order $p^t$. Cauchy theorem : (1) Let $G$ a ...
1
vote
2answers
24 views

Show that, for each finite $G$-set ($G$-action) on $X$, we have $|X| \equiv |X^G| \pmod n$

Let $G$ a group and $n \in \mathbb{Z}_{>0}$ an integers with the following properties. For each subgroup $H < G$ such that $H \not= G$, the integer $n \mid [G:H]$. Show that, for each ...
-2
votes
2answers
44 views

It is group or not? [closed]

Show that the set of vectors defined as directed line segments does not form a group (1) with respect to scalar product (2) with respect to vector product.
1
vote
1answer
35 views

Action of GL(2n,R) on set of linear complex structures

A linear complex structure on $\mathbb R^{2n}$ is an endomorphism $J: \mathbb R^{2n} \to \mathbb R^{2n}$ such that $J^2 = -Id$. (Then $J$ is necessarily an isomorphism.) We have an action of ...
0
votes
1answer
29 views

Action of permutation group on set of numbers is transitive

I would appreciate if someone could please tell their opinion about my proof. I think the proof makes sense, but I don't know if it's rigorous enough. Theorem: Let $S_n$ be a symmetric group of ...
0
votes
1answer
22 views

low-dim unitary groups and their actions

I need someone to explain for me the unitary groups $U(1)$, $U(2)$ and $U(3)$ and their actions: Specifically: $U(3)/U(2)$ $U(3)/U(2)\times U(1)$ $U(3)/U(1)\times U(1) \times U(1)$ I have seen ...
0
votes
0answers
29 views

If $G$ acts such that each nontrivial element fixes exactly $n$ points or none, then $|\Omega| \equiv n \pmod{|G_{\alpha}|}$

Let $G$ act faithfully such that $|\mbox{fix}(g)| \in \{0,n\}$ for each $g \ne 1$. Then $|\Omega| \equiv n \pmod{|G_{\alpha}|}$. This should be a corollary from two lemmata I will give; but I ...
-1
votes
1answer
72 views

$GL_2(\mathbb{Z}_2)$ acting on $\mathbb{Z}_2 \times \mathbb{Z}_2$ [closed]

This action induces the homomorphism $\phi:GL_2(\mathbb{Z}_2)\to S_4$, which is injective. Would it be correct to explicitly list the elements of $im(\phi)$ as the matrices in $GL_2(\mathbb{Z}_2)$?
1
vote
2answers
67 views

Homomorphism induced by group action

If $G$ is a group acting on $X$, which is a set of all left cosets of $H\leq G$ (such that $|G:H|=k$), by left multiplication, then this group action induces a homomorphism $\phi: G\to S_X$, such that ...
0
votes
2answers
65 views

If $G$ acts on $X$ then $\psi: X\to X$ is a bijection

I would appreciate if you could please express your opinion on this proof. I don't know how else this proof can be done. Theorem: If $G$ acts on $X$ then $\psi: X\to X$ defined by $\psi_g(x)=g\cdot ...
2
votes
1answer
41 views

Free circle action from a torus

So these lecture notes contain this statement (Exercise 3.3.5): If $T$ is a torus acting on a compact manifold $M$ such that every isotropy subgroup has codimension greater than one, then there ...
2
votes
2answers
95 views

counterexample showing that Maschke's Theorem does not hold if characteristic divides group order

I am taking a graduate Algebra course, and we were given the following example to see that Maschke's Theorem does not hold if the characteristic of the field F does divide the order of G: Let $F = ...
0
votes
1answer
20 views

Equidecomposability and Tarski's theorem

Suppose we have an action $G \curvearrowright X$. We say that two subsets $A, B \in X$ are equidecomposable (written $A \sim B$) if there exist a disjoint partition $(A_i)_{i=1}^n$ of $A$ and elements ...
0
votes
1answer
62 views

Understanding Groups Actions and Homomorphisms

I am trying to prove the following as exercises. Let the group $G$ act on the set $X$. We define the $kernel$ of this action as the normal subgroup $K = \{g \in G | \forall x \in X: g \cdot x = x\}$. ...
0
votes
1answer
36 views

Show that for transitive subgroup $A$, its centralizer divides $|\Omega| = |A : A_{\alpha}|$

Let $G$ be a group acting faithful on a finite set $\Omega$ and suppose the subgroup $A \le G$ acts transitive on $\Omega$. Then $|C_G(A)|$ divides $|\Omega|$, and if in additon $A$ is abelian, then ...
2
votes
1answer
88 views

Let $G$ act such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Then the Sylow $2$-subgroups acts regular on certain orbits

Let $G$ be a finite permutation group on $\Omega$ acting nonregular and transitive such that each nontrivial element fixes at most two points of $\Omega$. Suppose that for $\alpha \in \Omega$ the ...
1
vote
0answers
41 views

The alternating group $A_7$ cannot act transitive, nonregular and such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$ on any finite set

There is no finite set $\Omega$ on which $G \cong A_7$ acts transitive, nonregular and such that each nontrivial element has at most two fixed points. A little lemma first: Lemma: Let $G$ act ...
1
vote
0answers
25 views

Averaging measurable functions over actions of amenable groups

Let $G$ be a countable abelian group acting on a space $X$. It is known that such groups are amenable, i.e., there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$. (For finite groups this is ...
0
votes
0answers
31 views

Action of normal subgroup and relation of point stabilizers to normal subgroup

If $N \unlhd G$ and $G$ is a finite permutation group, then as the $N$-orbits form a system of blocks we have for each $\alpha \in \Omega$ that $G_{\alpha} \le G_{\{\alpha^N\}}$ and of course $N \le ...
1
vote
0answers
16 views

Direct caracterisation of certain subgroups of GL(E)

I'm wondering how to express certain subgroups of GL(E), E being a vector space of finite dimension n. These subgroup are caracterised as : let $ \sigma \in S_n $ , then $ G_{\sigma} $ is the ...
0
votes
0answers
37 views

The centralizer of an involution in a permutation group where $|\mbox{fix}(g)| \le 2$ for $g \ne 1$

Let $G$ be a finite permutation group which acts transitive and nonregular on $\Omega$ in such a way that each nontrivial element fixes at most two points. Let $\alpha, \beta \in \Omega$ be distinct ...
2
votes
0answers
52 views

Let $G$ act such that each nontrivial element has at most two fixed point and $P := O_p(N)$. Then $G_{\alpha}$ acts fixed point freely on $P$

Let $G$ be a finite permutation group acting nonregular and transitive such that each nontrivial element fixes at most two points. Lemma: (1) If $p$ is odd and divides the order of $G_{\alpha}$, then ...
1
vote
0answers
42 views

Reasoning about subgroups $H \le PSL(2,q)$ based on knowledge about subgroups of normalizers in it

Suppose that $G$ is a finite permutation group acting transitively and non-regularly on $\Omega$. Also suppose that each non-trivial element has at most two fixed points and $|\Omega| \ge 4$. Let me ...
1
vote
0answers
32 views

In a finite permutation group where $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Properties of orbit of $O_p(G)G_{\alpha}$.

Let $G$ be a finite permutation group acting transitive and non-regular on $\Omega$ with $|\Omega| \ge 4$. Suppose further that every nontrivial element has at most two fixed points. Now let ...
0
votes
1answer
27 views

Group of order $p^n$ acting on a set of set of order $pm$

Let $G$ be a finite group of order $p^n$ where p is prime. Suppose $G$ acts on a set $X$ of order $pm$ for some non negative integer $m$. Prove or disprove that if $F=\{x\in X : g.x=x \ \forall g\in ...
1
vote
1answer
35 views

Group Action and stabilizer problem.

$n$ is a positive integer the group $S_n$ acts on the set $A={1,2,....,n}$ as : $s\cdot i = s(i)$ for all $i \in $ {1,2,...,n}. We need to show that the action is faithful and the stabilizer $G_i$ ...
1
vote
2answers
33 views

Commutator Group of $S_4$ and $A_4$

This is directly out of Dummit and Foote S5.5 Q4. In class on Friday my professor gave this question to us as an exercise, with the answer being that all 3-cycles$\in S_4$ was the commutator group for ...
1
vote
1answer
38 views

Prove that if $G$ is finite, then any neighborhood of a $G$-invariant subset of $X$ contains a $G$- invariant neighborhood

Let $G$ be a group acting on a topological space $X$. Prove that if $G$ is finite, then any neighborhood of a $G$-invariant subset of $X$ contains a $G$-invariant neighborhood. I have no idea ...
2
votes
0answers
48 views

Group actions and cardinality of double cosets

I was recently asked this in my abstract algebra class on group actions which seems difficult for me and so need the help on: Let $ G $ be a group and $ H,K \leq G $ be two subgroups with the ...
1
vote
3answers
60 views

Prove that the binomial coefficent $\binom{2p}{p} $ is $\equiv 2\pmod{p}$ using group actions.

I have to prove that the binomial coefficent $\binom{2p}{p} $ is $\equiv 2\pmod{p}$ using group actions. I've tried with an action of $ C_p \times C_p$ upon the set of all numbers between $1$ and ...
1
vote
2answers
43 views

Group action of Symmetric group on arbitrary set A

I was recently given this problem in my Abstract Algebra course dealing with group actions, stating the following: Let A be a non-empty set and $ S_A $ is its symmetric group. Now assume we have a ...
2
votes
0answers
98 views

Proof that solvable permutation group whose fixed point set is restricted contains regular normal subgroup or Frobenius group on orbits

Let $p$ be a prime. Let $G$ be a solvable, non-regular, transitive permutation group such that some element fixes no point, and each element fixing some point fixes exactly $p$ points. Suppose that ...
3
votes
0answers
36 views

Free and proper action of a closed subgroup of a Lie group

I'm taking a course on Riemannian geometry and in my homework set I'm asked to prove that the (left) action of a closed subgroup $H$ of a Lie group $G$ on $G$ is free and proper. To prove that it is ...
4
votes
0answers
63 views

Is $D$ a metric on $X/G$ and does it induce the quotient topology?

Let $(X,d)$ be a compact metric space and $G$ be a finite group of homeomorphisms of $X$. Let $p:X\rightarrow X/G$ be the orbit map. Then we can define a (psuedo) metric on $X/G$ as follows - ...
4
votes
1answer
114 views

Quotient space $\mathbb{R}/\mathbb{Z}^{2}$ is not a manifold

I need to prove that $\mathbb{R}/\mathbb{Z}^{2}$ is not a manifold when $\mathbb{Z}^{2}$ acts (continuously) on $\mathbb{R}$ by $t\mapsto t+m+n\alpha$ where $\alpha$ is a fixed irrational for all ...
0
votes
0answers
25 views

Group actions producing an automorphism of a group formed from a power-set to itself and symmetric difference

Let the group $G$ act on the set $X$. For $g \in G$ and $A \in P(X)$ set $g.A = \{g.a \mid a \in A\} = \{x \in X \mid \exists a \in A: x = g.a \}$. this defines an action of $G$ on $P(x)$. I 'm ...
7
votes
1answer
77 views

Free $\mathbb{Z}_{2}$ action on the plane

Motivated by the following question we ask: Is there a free action of $\mathbb{Z}_{2}$ by homeomorphism on $\mathbb{R}^{2}$? Lie groups with no free $\mathbb{Z}/2\mathbb{Z}$ action