Questions tagged [group-actions]

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 $2n$ act 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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$3$-length of $(C_3\times C_3):GL(2,3)$ in GAP

I try to find $3$-length of the semidirect products group $(C_3\times C_3):GL(2,3)$. $p$-length means the number of factors in the shortest subnormal series which factors are $p$-groups or $p'$-groups....
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About a right action definition

Let's take this group action definition, taken from a textbook I'm reading: Given two groups $G$, $H$ and a homomorphism $\phi=G \mapsto \text{Aut}(H)$, we say G acts on H through $$h^\sigma=h^{\phi(\...
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2 votes
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A polynomial in 3 (commuting) variables which only gives 2 distinct polynomials (including itself) on permuting variables

This is a question from Oxford. I am stuck at last part. It says that every divisor of n! occurs when n=3 so that includes 2 but i cant find a polynomial in 3 (commuting) variables which only gives 2 ...
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5 votes
2 answers
178 views

Equivalent definitions of a group acting on a group?

I've always seen this definition of a group $G$ acting on a set $\Omega$, making this latter a $G$-set: Given a group $G$ and a set $\Omega$ we say a group acts on the set $\Omega$ when there's a ...
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1 vote
1 answer
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What is an invariant embedding?

Let $S$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $0$, equipped with a regular involution $\iota$, i.e., with an action of the order two (cyclic) group $...
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A question about moment maps

The following is from page 3 of the article Matsuki correspondence for sheaves: Let $G$ be a semi-simple Lie group with a maximal compact subgroup $K$. Let $\mathfrak{g}= \mathfrak{k} \oplus \...
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Group action on monoid with involution - Laws relating involution of element (from monoid) with inverse function (from group)

Preliminaries (Remarks on notation: To denote function application, I will use the Haskell notation $f\ x$, rather than the traditional mathematical notation $f (x)$. The expression $x^-$ denotes an ...
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1 answer
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Group action on a topological space and Haar integral (A question on a proof in a Palais' paper)

The paper is: R. Palais, "On the existence of slices for actions of non-compact Lie groups", Annals of Mathematics, vol. 73, 1961. In the proof of Proposision 1.2.6, p. 301, $X$ is a ...
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Let $G$ be a simple group which acts on $\Omega$. Let $\alpha \in \Omega$ such that $|O(\alpha)|=p$. Prove the order of $p$-sylow subgroup is $p$.

Let $G$ be a finite simple group which acts on $\Omega$. Let $\alpha \in \Omega$ such that $|O(\alpha)|=p$, ($O$ is the orbit of $\alpha$, $p$ is a prime number). Prove the order of $p$-sylow subgroup ...
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3 votes
1 answer
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Group action for signal

I am working through the book Geometric Deep Learning (https://arxiv.org/abs/2104.13478) and have hit the following problem (Chapter 3.1, page 14). We have a group $\mathfrak{G}$ and a set $\Omega$ ...
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The mean random walk on a spherical lattice excursion length

In this Jane Street puzzle, its solution says the mean stroll length is $20$. The arugment is kind of plausible but not clear. I would like to see a rigorous proof utilizing the symmetry.
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A question about group coset. [closed]

Let $G$ be a group, $H\leq G$. Denote an action $\alpha$ such that $G$ acts on $G/H, [G:H]=n.$ The action defined : $g_i.g_jH=(g_i \cdot g_j)H$ when $g_i\in G , g_jH$ is a coset in $G$. $$G=g_1H\cup ...
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How to Find Orbits and Stabilizers

The question is: Let $S$ denote the set of possible black-or-white colorings of the edges of an equilateral triangle. The triangle's symmetry group $D_6$ acts naturally on $S$. (a) How many orbits are ...
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1 vote
1 answer
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Power of $ 2 $ congruent to 1 mod n [closed]

Fix some odd $ n $. For which values of $ m $, $ 1\leq m \leq n $, is $ 2^m $ congruent to $ 1 $ mod $ n $? For $ n=3 $ the solutions are $ 2^2=4 \equiv 1 $ For $ n=5 $ the solutions are $ 2^4=16 \...
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2 votes
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Why does a fundamental domain have a freely acted upon element?

I want to understand the proof of Theorem 1.58 in "Groups, Graphs and Trees" by John Meier. I have some problems with the second line (see screenshot). With every connected simple graph $\...
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Prove new group action is properly defined. [closed]

In continuation with part (b) of the problem here. Show that there is injective map from the sets of actions on set $X$ to set of new-action on set $Y.$ How to proceed?
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Show that a Modified-group action, is also a group action.

Let $G$ be a group and $X$ a set. Define a modified-action as for which identity may not hold. Have a subset $Y$ of the set $ X$ where holds:$\,e\star x\mid x.$ Can it be assumed that $e\star y$ has ...
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1 vote
2 answers
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Intuitive meaning of transitive group action. [closed]

My idea (not read) is that a transitive group action enforces surjective mapping. The definition of transitive action is that there exists an $x \in X$ such that for any $y \in X$, there exists a $g \...
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3 votes
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Suppose a group $G$ splits as $A \ast_H B$ and $G > K \geq H$. Does $G$ also split over $K$?

Let $G$ be finitely generated, and consider a chain of subgroups $G > K \geq H$. We say that $G$ splits over $H$ if $G$ acts without inversions on a simplicial tree $T$ with some edge stabilised by ...
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1 vote
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Circles bounding $\mathrm{SL}(2,\mathbb{Z})$ fundamental domain translates

The (closure of the) "canonical" fundamental domain for $\mathrm{SL}(2,\mathbb{Z})\backslash\mathbb{H}$ is given by $F=\{z=x+iy\in\mathbb{H}:2x\in[-1,1],\lvert z\rvert\ge1\}$. $\mathrm{SL}(2,...
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Classification actions of finite groups

This is a very elementary question, but I'm having trouble finding anything explicit about it: Consider a finite group $G$ with an action on a finite set $X$. Is there a classification of such actions ...
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2 votes
1 answer
72 views

Find sizes of disjoint subsets to make transitive action by icosahedron on set $X$.

Could only prepare ground for this in my last post here, with a reduced group and set size. Kindly give hints, as sizes of subsets that can think of are too small: $X_3.$ Though read of tetrahedron ...
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Difference between group action and transitive group action. [closed]

These are my notes, request vetting and help in doubts. Made to understand problem stated at: Find sizes of disjoint subsets to make transitive action by icosahedron on set $X$. Let us start with an ...
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5 votes
1 answer
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How close is $ GL_n (2) $ to being $3$ transitive?

The group $ GL_n(2) $ acts transitively on the $ 2^n-1 $ nonzero vectors of $ \mathbb{F}_2^n $. This action is in fact 2-transitive since any pair of distinct nonzero vectors in $ \mathbb{F}_2^n $ is ...
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1 vote
1 answer
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Specific questions regarding the group action $G \times \mathcal{S}(G) \to \mathcal{S}(G)$

Problem statement: Let $G$ be any group and recall $\mathcal{S}(G)$ denotes the set of subgroups of $G$. Let $G \times \mathcal{S}(G) \to \mathcal{S}(G)$ be given by $$g \cdot H = gHg^{-1}$$ Prove ...
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1 vote
1 answer
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Show that for the coset action $G \times G/H \to G/H$ the stabilizer of a coset is the conjugate of $H$ by $a$

Problem Statement: Let $G$ be a group and $H<G$ a subgroup. Consider the coset action $G \times G/H \to G/H$ given by $g \cdot (aH) = gaH$ for all $g,a \in G$. prove that $$\text{stab}_G(aH) = aHa^...
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1 vote
1 answer
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How many distinct transitive actions does quaternion group have.

Quaternion group $Q_8$ has though $6$ elements of order $4$, but there are three sets ($(i, -i), (j, -j), (k, -k)$) having seperate subgroups, with identity and $-1$ common only. Apart from that have ...
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1 answer
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How many distinct transitive actions exis in a cyclic group of order $n.$

The number of distinct transitive actions should depend on the number of orders possible for $g\in G= C_n$. Say, take $n=12:$ Hence, the answer as per Lagrange's theorem should be: $6$, as orders of $...
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2 votes
0 answers
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Generalised Symmetric group and Semidirect Product

$G= \mathbb{Z}_r \wr \mathfrak{S}_n:=(\mathbb{Z}_r)^n\rtimes\mathfrak{S}_n$ is the generalised symmetric group where the element $G$ is denoted by $(f,\pi)$ where $f:\{1,2,\dots n\}\to \mathbb{Z}_r$ (...
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Closed embedding induces closed embedding between balanced products?

Let $G$ be a finite group, $H\subset G$ a subgroup and $f: X\to Y$ an $H$-equivariant continuous map between topological spaces endowed with an $H$-action. Define the balanced product $$ G\times_H X :=...
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1 answer
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Compute the number of distinct actions of cyclic group $C_n$ on a set $X,$ s.t $|X|= n+1.$

There are $n+1C_n = n+1$ combinations possible, with $n!/n= (n-1)! $ orderings possible in each; leading to a total of $(n+1).(n-1)!$. Let $n= 6.$
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1 answer
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Number of different ways to get permutations of disjoint cycles of given length.

This question is derived (as want to derive the formula for below problem, and also general approach) from : Compute the number of distinct actions of $C_m$ on set $X,$ s.t. $|X|= n= 2m+1.$ Let ...
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5 votes
1 answer
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When is multiplication by an element from a transformation group bijective?

I am reading Allan Clark's Elements of Abstract Algebra and he states the following about transformation groups. We note that each $g \in G$ determines a one-to-one correspondence $g: X \to X$ given ...
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3 votes
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Actions on Bernoulli space have almost the same orbits

Let $(\mathbb{B}, \nu)$ be the binary space with probability measure $\nu(0) = \nu(1) = \frac{1}{2}$. The map $T : \mathbb{B}^{\mathbb{N}} \to \mathbb{B}^{\mathbb{N}}$ is defined as left addition with ...
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Let a cyclic group $A=\langle a\rangle$ act on a group $G$ whose order is odd. If $[G,a^2]=1$, then $\{x\in G: x=x^{-a}\}=\{[x,a]:x\in G\}.$

Let a cyclic group $A=\langle a\rangle$ act on a group $G$ whose order is odd. If $[G,a^2]=1$, then $$\{x\in G: x=x^{-a}\}=\{[x,a]:x\in G\}.$$ My attempt: Since $[G,a^2]=1$, so $x^{a^2}=x$ for all $x\...
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Invariant differential form and compact Lie group actions

Let $G$ be a compact Lie group acting on a manifold $M$. Denote the action by $\tau_g(p):=g\cdot p$. Let $dg=\nu$ be a left invariant volume form on $G$ such that $\int_G \nu =1$. Let $\alpha$ be a ...
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How to classify groups of order $6p$ where $p$ is a prime and of the form $p=6\ell+1$, especially $|G|=2022$?

How to classify groups of order $6p$ where $p$ is a prime and of the form $p=6\ell+1$, especially $|G|=2022$? I found some lists like this one and this one, both show that $|G|=6p$ divides into two ...
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1 answer
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Why $\mu_3 = \operatorname{ diag} (\zeta_3, \zeta_3)$ is the kernel of the G'-action?

Here is the paper I am trying to understand: But I do not understand why $\mu_3 = \operatorname{ diag} (\zeta_3, \zeta_3)$ is the kernel of the G'-action? could anyone show me the details of the ...
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8 votes
1 answer
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Proving an identity regarding character of irreducible representation

Let $\frak{X}$ be an irreducible representation of a finite group $G$ affording the character $\chi$. Prove that for every $x,y\in G$: $$\chi(x)\chi(y)=\frac{\chi(1)}{|G|}\sum_{z\in G}\chi(yzxz^{-1}).$...
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1 vote
1 answer
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Finding stabilizer and orbit of power set of a group

Let $H$ be a subgroup of a finite group $G$. Let $A$, $B$ $\in \mathcal{P}(G)$. Define $A$ to be conjugate to $B$ with respect to $H$ if $B=hAh^{-1}$ for some $h\in H$. Then find orbit and stabilizer ...
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2 votes
2 answers
108 views

When a group action is transitive, is it for all elements of the group acting on the set?

I know that a group action is transitive when there is one orbit. Say that $G$ is a group acting on the set $A$. The identity element of $G$ will clearly create $|A|$-many orbits. But the other ...
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-4 votes
1 answer
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What condition in the definition of group action makes g.(pq) = (g.p)(g.q)? [closed]

Given the set $P$ of polynomials of degree $3$ in $2$ variables. I revised the definition of group action on $P$, but I do not see any reason in the definition that makes us say that $g.(pq) = (g.p)(g....
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3 votes
0 answers
31 views

Notion of $G$-stable subsets in Moerdijk and Mrčun's Introduction to foliations and Lie groupoids

This question is related to this this older question. Let $M$ be a smooth manifold and $G$ be a group of diffeomorphisms of $M$. In page 35 of Introduction to Lie foliations and Lie groupoids, ...
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Characterizing restriction of scalars, recognizing matrices for the action of multiplication in a field extension

Let $E/F$ be a finite field extension of degree $[E:F]=n$. Then multiplication by $e\in E$ is linear over $F$ and below we consider the units $E^{\times}$ as a subgroup of the general linear group of ...
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-2 votes
0 answers
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What does $G/H$ act mean?

Let $G$ be a group and $H$ is normal subgroup of $G$. Let $G$ acts on set $V$. In this situation, in general, what does $G/H$ acts on $V$? $G/H\times V\to V$,$ (g\mod{H},v)\to gv$ contains no ...
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3 votes
0 answers
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Does every effective $T^4$-action on a simply connected 6-manifold admit a free subtorus action?

All simply connected $6$-manifolds which admit effective $T^4$-actions have been classified by Hae Soo Oh (See Theorem 1.1). The classification shows that only the following simply connected $6$-...
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2 votes
1 answer
49 views

Is there a group action which is not free but induces covering map?

Let group G act on topological space X, i.e. there is a group homomorphism from G to the homeomorphism group of X. As we all know, if this group action is properly discontinuous, then it is free and ...
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1 vote
3 answers
69 views

Left Translation Action is Ergodic with respect to Haar

I am trying to solve exercises on ergodic group actions, from the A. Ioana's lecture notes "Orbit Equivalence of Ergodic Group Actions". The following exercise (p.3, Exr.1.14) has two parts, ...
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2 votes
1 answer
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Left action and group automorphism

I know a group acts on itself by conjugation forming its inner automorphism group. At the same a group acts transitively onto itself by the left action, forming the basis for the Cayley theorem. I was ...
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6 votes
1 answer
196 views

For the set $X=\{g\in G:g^p=1\}$, show that $p$ divides $|X|$.

Let $G$ be a finite group and $p$ be a prime divisor of $|G|$. Consider the set $X=\{g\in G:g^p=1\}$. Show that $p$ divides $|X|$. My attempt: Consider the action of $G$ on $X$ by conjugation. Then ${...
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