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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1answer
30 views

If $A$ is $G$ - invariant then so is $\bar A$

Let $G$ be a topological group acting continuously on a topological space $X$. Let $A$ be a $G$ - invariant subspace of $X$. Then is it true that $\bar A$ is also $G$ - invariant? (where $\bar A$ is ...
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2answers
43 views

kernel of action.

If $\phi : G \to Perm(G/H)$ where $\phi$ is the group action on $G/H$ by $G$. $\phi := g(g'H) = (gg')H$ Why is the kernel of $\phi$ equal to $\cap_{x\in G} xHx^{-1}$ I thought the kernel is $\ker ...
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0answers
32 views

If $G$ acts so that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$. Conditions such $S \in \mbox{Syl}_p(G)$ has maximal class.

Let $G$ be a nonregular, transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Further suppose that for $g ...
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0answers
17 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ maximal, why is $N_M(M_{\alpha}) \in \mbox{Syl}_p(M)$?

Let $p$ be an odd prime. Suppose $G$ is solvable and acts as a nonregular and transitive permutation group on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points. ...
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0answers
36 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, $M$ maximal with $|G : M| = p$, then $|M / L| = p$ for semiregular $L \unlhd G$.

Let $G$ be a solvable, nonregular and transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. And suppose that ...
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0answers
19 views

The uniqueness of the Frobenius representation as handled in textbooks, for example Kurzweil/Stellmacher

As written on Wikipedia:Frobenius_group The Frobenius kernel $K$ is uniquely determined by $G$ as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by ...
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0answers
26 views

Group of rotational symmetries of regular tetrahedron is isomorphic to $A_4$

Let $G$ be the group of rotational symmetries of a regular tetrahedron. I'm trying to think of an argument proving that since $G\cong H$, where $H\leq S_4$, $H=A_4$. There are 12 rotational ...
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0answers
81 views

Normal subgroup $H$ of $G$ with same orbits of action on $X$.

I have a somewhat broad question related to group actions and their restriction to a normal subgroup. If we have a group action $\sigma : G \times X \rightarrow X$ with orbits $G_x$, and a normal ...
1
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1answer
21 views

Group actions - modulo 4

I am having a bit trouble understanding group actions. if I am given a set A = {a,b,c,d} and a group action s: Z mod 4 -> $S_A$, how would one then be able to show if there exists a group action s ...
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2answers
165 views

group actions on spheres

Let $\mathbb{Z}/2$ act on the $m$-sphere $S^m$ freely and properly discontinuously. If the action is not trivial, can we conclude that the action is homotopy equivalent to the antipodal action? That ...
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1answer
62 views

Transitive actions on sets.

Does there exist a transitive action of $S_4$ on the set $\{1,2,3,4,5\}$ ? I would say no, because the cardinality of our set is bigger than $4$, but I am not sure how to prove this. My suggestion ...
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0answers
38 views

Moment map in general

Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if ...
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2answers
37 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then every element outside of $N$ fixes at most $p$ $N$-orbits.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Also assume that for $g \notin ...
2
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1answer
27 views

The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$

Let $G$ be a permutation group acting transitively on $\Omega$ and suppose $N \unlhd G$ is a normal subgroup of $G$. Assume that for $g \in N_g(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\alpha}^g = ...
2
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1answer
22 views

Sign convention when commuting shifts and tensor product

In Markl, Schnider, and Stasheff's Operads in algebra, topology, and physics, they give the observation $\mathfrak{s}^{-1} \mathcal{E}nd_V \cong \mathcal{E}nd_{V[-1]}$ (lemma 3.16) as motivation for ...
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1answer
33 views

The kernel of an action on blocks, specifically the action on the orbits of normal subgroup

Let $G$ be a permutation group acting transitively on some set $\Omega$ and suppose we have a normal subgroup $N \unlhd G$. Then the orbits of $N$ form a system of blocks, and if $\Delta$ is such an ...
0
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0answers
26 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G_{\alpha}N$ is normal if we have $p$ orbits of $N$.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either has no fixed point or exactly $p$ fixed points. Suppose that for $g \notin N_G(G_{\alpha})$ we ...
2
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0answers
35 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ is maximal normal. Then $|G/M| = p$.

Let $G$ be a transitive permutation group on $\Omega$ which fulfills the following property (P) (P) each nontrivial element fixes no point or exactly $p$ points. for some odd prime $p$. Further ...
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0answers
29 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 ...
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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
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0answers
30 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
31 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 ...
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0answers
31 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
31 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 ...
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0answers
49 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
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0answers
21 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
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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
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1answer
54 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 ...
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0answers
32 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
0answers
28 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
18 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 ...
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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 ...
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2answers
41 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
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1answer
33 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 ...
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1answer
25 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
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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 ...
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0answers
28 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 ...
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1answer
70 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 $GL_2(\mathbb{Z}_2)$?
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2answers
51 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 ...
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2answers
64 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
39 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
36 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 = ...
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1answer
19 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
56 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
31 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
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
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0answers
31 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
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0answers
17 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 ...