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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2
votes
1answer
78 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
32 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
20 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
30 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
15 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
32 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
47 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
36 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
29 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
25 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
31 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
30 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
39 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
34 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
91 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
29 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
59 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 - ...
3
votes
1answer
101 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
21 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
75 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
4
votes
0answers
46 views

Can we extend the group action from subgroups?

Let $G$ be a group and $H,K$ be subgroups of $G$ such that $G=<H,K>$. Suppose that $H,K$ acts on the set $S$. Is there any condition that which guarantees that action of $H,K$ is extended to ...
2
votes
1answer
68 views

Show that maximal abelian normal subgroup of $p$-group contains the commutator subgroup

Let $P$ be a $p$-group and $A$ a maximal abelian normal subgroup of $P$. If $|A : C_A(x)| \le p$ for all $x \in P$, then $P' \le A$. As far as I know I have no idea how to bring the condition ...
6
votes
1answer
148 views

Is the action of a finite group always discontinuous?

Let $G$ be a finite group acting on a topological space $X$. Is it true that the action of $G$ is always proper? To say that it is proper we have to show that the map $\theta : G \times X ...
0
votes
0answers
47 views

For $U := G_{\alpha}\cap G_{\beta}$ there exists an involution in $N_G(U) \setminus U$ when the number of fixed points in $G$ is bounded by $2$

Let $G$ be a finite, transitive, faithful group acting on $\Omega$ such that each nontrivial element has at most two fixed points. Lemma: If $\alpha \in \Omega$ and $1 \ne X \le G_{\alpha}$, then ...
2
votes
0answers
37 views

A problem of a discrete group of smooth isometries acting discontinuously on a smooth manifold.

Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly ...
5
votes
1answer
52 views

What is the closure of $S^1\times \mathbb{Z}$ in Homeo$(S^2\times \mathbb{R})$?

Let $X = S^2 \times \mathbb{R}$ and $G=S^1\times \mathbb{Z}$. Let $G$ act on $X$ as follows - $S^1$ acts on $S^2$ by rotation leaving the north and south poles fixed and acts trivially on ...
3
votes
0answers
73 views

Exercises on group theory [closed]

What are some difficult, challenging and fair exercises in group theory? I know it is quite general, in particular I am referring to these areas of group theory: theory of automorphism group ...
1
vote
0answers
58 views

Steps in proof that $S \in \mbox{Syl}_2(G)$ implies $G$ has a normal (Frobenius-) subgroup of index $2$ under fixed points restrictions

Suppose $G$ is a finite, non-regular, faithful, transitive group such that each nontrival element has at most two fixed points. Suppose $S$ is a Sylow-$2$-subgroup of $G$ and $\alpha \in \Omega$. If ...
1
vote
2answers
54 views

Stabilizers, Orbits, and the Symmetric Group

I have a homework question I am a bit confused on. Let $S_n$ act on the set of ordered pairs $(i,j)$, $1\leq i,j \leq n$ such that $\omega \cdot (i,j)= (\omega i,\omega j)$. I have already proved ...
2
votes
1answer
59 views

Properties of point stabilizers of $PSL(2, q)$

Let $G$ be a transitive, non-regular finite permutation group such that each non-trivial element fixing some point fixes exactly two points. Suppose that $G \cong PSL(2, q), q > 5$ and $H = ...
4
votes
1answer
99 views

Properties of group acting such that each non-trivial element fixes no point or exactly $p$ points

Let $p$ be a prime and $G$ a faithful non-regular transitive finite group acting on $\Omega$ with $|\Omega| > p$ such that some element fixes no point, and that each nontrivial element fixing ...
6
votes
0answers
46 views

Proving that the given map has the path lifting property

Let $X$ be a locally compact metric space and $G$ be a discontinuous group of homeomorphisms of $X$. I need to show that the orbit map $p :$ $X \rightarrow X/G$ has the path lifting property. ...
0
votes
0answers
37 views

If $G$ acts such that each non-trivial element either has no fixed point or exactly two, then there exists fixed-point free involutory map on $\Omega$

If a finite group $G$ acts on a set $\Omega$ non-regulary (i.e. there is some element fixing some point) and each element having some fixed point has exactly $n$ fixed points, then we say the group is ...
0
votes
0answers
52 views

Theorem about non-regular group action, where an element fixes no points or exactly $p$ points

I have a question on the following proof. All groups are assumed to be finite. But first I will mention a lemmata: Lemma: Let $G$ act faithfully and non-regular as a group such that there exists some ...
3
votes
1answer
23 views

If $G\curvearrowright X$ and $H\leq G$ then $\bar{H}x = \overline{Hx}$

If $G$ is a topological group acting effectively on a topological space $X$ and $H$ is a subgroup of $G$ then is it true that $\overline{H}x = \overline{Hx}$, where $\overline{H}$ is the closure of ...
5
votes
1answer
58 views

If $G$ is a group of isometries of $X$ then prove that $X/G$ and $X/\bar{G}$ are homotopically equvalent

Let $X$ be a connected, locally path connected, locally compact metric space. Let $G$ be a group of isometries of $X$ (that is a group of homeomorphisms of $X$ with itself that preserves distance). ...
2
votes
1answer
43 views

Group Theory, conjugation of permutation group

I've been given the following question in the context of group actions through conjugation but I'm having difficulty understanding what is being asked Let $\tau$ be any permutation in $S_m$. Let ...
0
votes
1answer
26 views

Bounding the index of subgroup intersection

Full disclosure: this is a homework problem, but it is not assigned to turn in for credit. The problem is from Dummit and Foote, Chapter 3.2: Suppose $H, K$ are subgroups of finite index a group (not ...
3
votes
0answers
39 views

Topology on $\text{Homeo}(X)$ Which Captures Topological Group Actions.

Definition. Let $G$ be a group and $X$ be any set. We may define a group action of $G$ on $X$ as map $\cdot: G\times X\to X$ such that $e\cdot x=x$ for all $x\in X$ and $g\cdot(h\cdot x)=gh\cdot x$ ...
3
votes
1answer
65 views

What is the closure of $\mathbb{Q}$ in Homeo$(\mathbb{R})$?

I read in a paper that if I consider the group of rationals, $\mathbb{Q}$ acting on the real line, $\mathbb{R}$ by addition then taking the closure of $\mathbb{Q}$ in Homeo$(\mathbb{R})$ gives a copy ...
1
vote
1answer
51 views

Quotient of $\mathbb{P}^n\times\cdots\times \mathbb{P}^n$ by $S_d$: is it projective?

Let us have $\mathbb{P}^n\times\cdots\times \mathbb{P}^n$ ($d$ copies of $\mathbb{P}^n$), and we have the symmetric group $S_d$ act by permuting the factors. Is the quotient projective? I have the ...
1
vote
1answer
34 views

Constructing a group which has fixed-point free element and element with exactly $n$ fixed points

Given some $n \ge 1$, how could we construct groups $G$ such that there exists some non-trivial element fixing no points and some non-trivial element fixing exactly $n$ points. If given a transitive ...
4
votes
2answers
99 views

Proper and discontinuous action of a group

I came across the following problem of algebraic topology that I couldn't solve. Let $\Gamma$ be a group which acts properly and discontinuously in a topological Haussdorf space $X$. Let $H ...
0
votes
0answers
34 views

Number of $p$-Sylow subgroups in $D_{2n}$

Let $2n = 2^ak$, where $k$ is odd. We wish to show that the number of $2$-Sylow subgroups of $D_{2n}$ is $k$. My approach has been to construct such a $2$-Sylow subgroup $P_2$, and then show that the ...
2
votes
0answers
39 views

Alternative solution to group theory problem

Exercise Let $G$ be a $p$-group and $g \neq H \leq G$. Prove $\exists g\in G\setminus H$ such that $$gHg^{-1}=H$$ I saw a solution to these exercise using induction on $n$ where $p^n$ is the order of ...
1
vote
0answers
36 views

Is this map well defined?

Let $X$ be a connected, locally path connected, locally compact metric space and $G$ be a group of homeomorphisms of $X$. Let $\bar{G}$ be the closure of $G$ in Homeo$(X)$ endowed with the compact ...
2
votes
1answer
50 views

Question about proof of orbit-stabilizer theorem

I am reading the proof here: https://proofwiki.org/wiki/Orbit-Stabilizer_Theorem In the last lines, it says that $\text{Orb}(x)$ has the same number of elements as $G / \text{Stab} (x)$, So ...
2
votes
1answer
42 views

Nonabelian group of order $pq $

I just showed that if $G $ is a nonabelian group of order $pq $, $p <q $, then it has a non normal subgroup $K $ of index $q $. But now I want to show that $G $ is isomorphic to a subgroup of the ...