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
17 views

How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...
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2answers
18 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
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1answer
74 views

Categorical Quotient and group actions

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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3answers
22 views

Misunderstanding the definition of a cycle (cyclic permutation)

Given a set $E$, let $\mathfrak{S}_E$ be the group of permutations of $E$. Definition.$\ \ $ Let $E$ be a finite set, $\zeta$ a permutation of $E$, and $\overline{\zeta}$ the subgroup of $\...
2
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1answer
31 views

The projective space as a homogeneous space

I want to understand why the projective space $\mathbb RP^n$ is diffeomorophic to $SO(n+1)/O(n)$? and why we can write the latter as $O(n+1)/O(n)\times O(1)$?
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29 views

Explicit Dehn twist for $S^n\times S^n$

Fix $n\neq 2$ and let $M=S^n\times S^n$. The diffeomorphism group Diff($M$) acts on the homology group $H_n(M)\simeq \mathbb Z^2$ inducing a surjection $d: \text{Diff}(M) \rightarrow \text{SL}(2,\...
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1answer
29 views

Showing that $x$ is an element of group $G$ by left multiplication

$G$ is a group and $H \leq G$ with $|G:H|=3$. Show that $x$ is an element of $H$ if $x \in G$ with $|x|=7$. Hint: let $\langle x \rangle$ act on $G/H$ by left multiplication and look at the orbits. ...
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0answers
39 views

Do finite groups act admissibly on separated schemes of finite type over k

Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$...
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3answers
80 views

An erroneous application of the Counting Theorem to a regular hexagon?

I'm trying to count the unique orbits of a regular hexagon such that each vertex is either Black or White and each edge is either Red, Gree, or Blue. The group I've chosen to act on the hexagon is the ...
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1answer
36 views

In transitive (non-trivial) group action, there must be at least one group element without fixed point

Let a finite group $G$ act transitively on a finite set $S$ with $|S| \geq 2$. The problem is to show that not every $g \in G$ can have a fixed point in this action. I proved this on my own, but I'm ...
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1answer
27 views

Interpretation of the join of two stabilizer subgroups

Let $G$ be the group acting on two sets $X, Y$. Let $G_x$ and $G_y$ be stabilizer subgroups of some elements $x \in X, y \in Y$. It is easy to see that $G_x \cap G_y = G_{(x,y)}$, when we combine two ...
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0answers
19 views

Is $Y/K$ homeomorphic to $Y'$ as defined below -

Let $G$ be a topological group acting on a topological space $X$ in such a way that there are only finitely many orbits. We will fix points $x_1,\cdots,x_n\in X$ and let $X=\bigcup_{i=1}^n G\cdot x_i$ ...
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1answer
58 views

Categorical Quotients and Group Actions on Varities

So I am given that Let $G = Z/dZ$ where d ≥ 1. Let w be a generator for G and let G act on $A^ {n+1}$ via $w(x_{0}, . . . , x_{n})$ = $(wx_{0}, . . . , wx_{n})$. How can I Show that the ...
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0answers
27 views

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$?

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$? This question is inspired by noting that if we have a Hamiltonian Lie group action $G\curvearrowright (M,\...
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1answer
38 views

If a subgroup has smallest prime index, then it is normal

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...
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2answers
48 views

How is the kernel of a group action defined?

Question: Show that the kernel of the group action of $G$ acting on set $A$ is equal to the kernel of the corresponding permutation representation of this action. I'm lost in this definition as ...
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1answer
40 views

Finite groups and prime divisors. Understanding how to deduce a claim from a certain proof.

In my algebra textbook, it goes like this. First, there is presented Cauchy's theorem: Let $G$ be a finite group, and let $p$ be a prime divisor of $|G|$. Then $G$ contains an element of order $p$....
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0answers
37 views

What does “factoring out an (group) action $\tau$ of a group $G$ acting on some set $E$” mean?

I am reading a survey article where they define the following objects: $\Gamma:=\mathbb{Z}^{n}$ seen as a group of translations. $\mathbb{T}:=\mathbb{R}^{n}/\Gamma$ is the $n$-dimensional ...
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1answer
16 views

Is it true that if all $G^\circ$ - orbits are closed in $X$ then all $G$ - orbits are closed in $X$?

Let $G$ be a Lie group acting continuously on a topological space $X$. Let $G^\circ$ be the connected component of the identity element of $G$ and let $[G:G^\circ]$ be finite. Then is the following ...
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3answers
53 views

G acts on X transitively, then there exists some element that does not have any fixed points

Let $X$ be a transitive $G$-set. ($G$ acts on $X$ transitively.) If $X$ is finite and has at least two elements, show that there is some element $g$ $\in$ G which does not have any fixed points; that ...
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1answer
34 views

The number of different G-actions on X [closed]

Let $X$ $=$ $\{$$1$, $2$, $3$$\}$ and $G$ $=$ $\mathbb Z_2$. How many different G-actions are there on $X$? Just learned group action. Need some hint on this one. Thanks.
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1answer
29 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
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1answer
53 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
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2answers
48 views

If $\sigma \in S_n$ has order some prime $p$, then is $|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p$? [closed]

Let $\sigma \in S_n$ be such that $o(\sigma)=p$ (some prime). Then is it true that $$|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p\ ?$$
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23 views

Every orbit $G\cdot x$ nonmeager is Baire.

For proof of Effros Theorem I have that $G$ is a Polish group and $X$ is a $G-$space Polish, but I need to show that if the orbit $G\cdot x$ is nonmeager then $G\cdot x$ is Baire in its relative ...
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0answers
78 views

Group action and orbit space

Suppose some group, G, acts on a space, X. Then an orbit of some $x\in X$ is defined as $$G.x = \lbrace g.x \mid g\in G\rbrace$$ Now consider the orbit space, $X/G$, the set of all orbits. I'm finding ...
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1answer
98 views

Is $(\mathbb R^3\setminus \{0\})/\mathbb R^*$ a smooth manifold?

Let $G=\mathbb R^*$ act on $X=\mathbb R^3\setminus\{0\}$ by pointwise multiplication. That is for any $t\in\mathbb G$ and $(x_1,x_2,x_3)\in X$ we have $$t\cdot(x_1,x_2,x_3)=(tx_1,tx_2,tx_3)$$ Is ...
4
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2answers
33 views

Show that giving a right-action of a group $G$ on a set $A$ is the same as giving a left-action of $G^{op}$ on A

This is a part of an exercise from "Algebra: Chapter 0" by Paolo Aluffi. First, I provide the necessary definitions. An action of a group $G$ on a set $A$ is a set-function $\rho: G \times A ...
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1answer
30 views

References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
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0answers
35 views

Fixed point set as an inverse limit

We can regard a group action on a set as a functor $$ F: BG \to Set\;, $$ where $BG$ is the category with one object and a morphism for each element of $G$, and $Set$ the category of sets. Now, is ...
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0answers
16 views

connected or compact Dirichlet domain

Let $G$ be a second-countable locally compact group and $d$ be a proper (i.e. bounded closed sets are compact) left invariant metric on $G$. Let $\Gamma$ be a lattice subgroup of $G$. Consider the ...
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1answer
31 views

Group extension that doesn't realize a coupling

Let $E$ be an extension of $N$ by $G$: $$N \hookrightarrow E \twoheadrightarrow G$$ If $N$ is abelian, then $E$ uniquely defines an action of $G$ on $N$. More generally, it defines a unique class $\...
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0answers
42 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
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0answers
24 views

Let $G$ be a $p$-group, with $|G|= p^n$. Show that $G$ has a normal subgroup of order $p^m$ for each integer $0 < m < n$. [duplicate]

I think I have solved a problem using one of the sylow theorems. But, if this proof is correct, I think I've cheated a little. Since the chapter on Sylow theorems comes directly after the chapter on ...
1
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1answer
27 views

Non-zero fixed point of some linear action on any finite group

Let $G$ be a group , $F$ be a field , $n$ be a positive integer , a map $h:G \times F^n \to F^n$ is called a linear action if there is a group homomorphism $f:G \to GL(n,F)$ such that $h(g,v)=f(g)v,\...
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1answer
43 views

$GL_2(\Bbb C)$ acts on a certain set

Let $G:=GL_2(\Bbb C)$, $B$ and $T$ be the subgroup consisting of all upper triangular and diagonal matrices in $G$, respectively. Set $w:= \left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\...
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2answers
53 views

The orbit of a compact Lie group action

Let $G$ be a compact Lie group acting on a manifold $M$. For each $p\in M$, we define the orbit of $p$ as $G\cdot p:=\{g\cdot p: g\in G\}$. The isotropy group of $p$ is $G_{p}=\{g\in G:g\cdot p = p\}$....
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47 views

Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
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1answer
25 views

a basic question in crossed product for compact group action

I am quite new into crosssed product of Fréchet algebras or C$^*$-algebras. So if the question is too basic please excuse me. Suppose we have two Fréchet algebras or C$^*$-algebras $A$ and $B$ and ...
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11 views

Representatives of the conjugacy classes of s5 [duplicate]

List the partitions of 5 and corresponding representatives of conjugacy classes in s5. What is the procedure to find the representatives of the conjugacy classes?
3
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1answer
33 views

Example of a free group action that is not proper.

I have been trying to think about Lie group actions on smooth manifolds and what the quotient spaces look like. I have a proof that compact Lie groups produce proper actions on manifolds, as well as ...
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0answers
85 views

Dense and turbulent orbits

In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition ...
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1answer
23 views

Subsets of $G$-sets with sharply transitive $G$-action

Let $G$ be an infinite group acting sharply transitively on a set $X$. Let $Y\subset X$ be a proper subset. Is there a subgroup $H\leq G$ which acts sharply transitively on $Y$ ? I think this is ...
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0answers
22 views

Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description

What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ...
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1answer
60 views

A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n $

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
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0answers
50 views

Show that $S^1$ acts on $S^3$

$S^3=\{(z_1, z_2) \in \mathbb{C^2} \mid |z_1|^2 + |z_2|^2 = 1 \}$ Show that $S^1$ acts on $S^3$ by $z \cdot (z_1, z_2)=(zz_1, zz_2)$ An action of a topological group $G$ on a topological ...
5
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1answer
63 views

Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ $\mathbb{g}=\begin{...
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0answers
17 views

circle actions on spheres

I'm considering the following action of $S^1$ on $S^3$: $$ e^{i\theta}.(z_1,z_2)=(e^{i\theta}z_1,e^{iq\theta}z_2) $$ It is clear that when $q=1$ the quotient space is $S^2$. Is there any description ...
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2answers
22 views

Group action decomposes $X$ into distinct orbits

Define the group action as $g\cdot x:=g^{-1}xg.$ Let $G=A_5$, and $X=\{\sigma\in A_5:=\sigma=(a,b,c,d,e)\}.$ Show that the group action on X decomposes $X$ into two distinct orbits. There are 60 ...
0
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1answer
36 views

Group action and equivalence relation

Let $G$ be finite, and group action on $X\subseteq G$: $g\cdot x:=g^{-1}xg$. Let $G=S_n$, and $X=S_n.$ Show that $[x]_R$ consists of all elements of $S_n$ that are of the same cycle-type as $x$. I ...