5 votes

Intuitive meaning of transitive group action.

Let us consider a group action of $G$ on a set $X$ defined by $(g, x) =g\star x$ Let $x\in X$ , then orbit of $x$ is defined as $\mathcal{O}_x=\{g\star x:g\in G\}$ Claim: The relation $\sim$ on $X$ ...
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Proving an identity regarding character of irreducible representation

Theorem Let $\chi \in Irr(G)$, and $x,y \in G$, then $$\chi(x)\chi(y)=\frac{\chi(1)}{|G|}\sum_{z \in G}\chi(xy^z)$$ Before proving this theorem we need to set notation and an observation. Write $\...
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4 votes
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Group action for signal

For clarity, it should be made clear that $\mathcal{X}(\Omega)$ is not a signal, but the vector space of all signals $x\colon\Omega\to\mathcal{C}$ to some fixed space $\mathcal{C}$ which is suppressed ...
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4 votes
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When is multiplication by an element from a transformation group bijective?

Consider the trivial group action, defined by $g\cdot x=x$ for every $x\in X$. It's not effective. For a less trivial example, consider a non-trivial normal subgroup $H\triangleleft G$. Let $G$ ...
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For the set $X=\{g\in G:g^p=1\}$, show that $p$ divides $|X|$.

$X$ comprises all and only the elements of $G$ of order $p$, plus the identity. Such elements are grouped in $m$ trivially intersecting subgroups each of order $p$. Therefore: \begin{alignat}{1} |X| &...
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Intuitive meaning of transitive group action.

First, notice that orbits partition the given set into disjoint subsets. That is, any two orbits are either same or disjoint, and the union of all the orbits is the whole space. Now, suppose $O_1$ and ...
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Number of different ways to get permutations of disjoint cycles of given length.

Remember that an action is a homomorphism from $C_3$ into $S_7$, here. (A group action on a set $X$ always gives a homomorphism from $G$ into $\rm{Sym}(X)$.) The $3$ inequivalent actions are $1\to e,\...
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How many transitive actions does the quaternion group have, up to isomorphism?

Hint Just like in your previous post about transitive actions of $C_{12}$, the general result (that @Derek Holt clued us in on) is that there are as many inequivalent actions as conjugacy classes of ...
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2 votes

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

Elements of $G$ do not have orbits. Elements of $A$ have orbits (and orbits are subsets of $A$). If a group $G$ is acting on a set $A$ and $a \in A$, then we denote the orbit of $a$ by $\operatorname{...
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How many transitive actions does $C_{12}$ have, counted up to isomorphism?

That there's only $6$ conjugacy classes of subgroups in $C_{12}$ implies that there's only $6$ possible transitive actions. See this answer.
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Left action and group automorphism

To esentialy sum up and promote the comment stream to the question itself to an answer: Let the group under discussion be denoted by $G$. The left (or right) action by $a \in G$ given by left (right) ...
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Classification actions of finite groups

I would say that we can first decompose $X$ into invariant subsets. By "invariant" subset do you mean that $G\cdot A\subseteq A$? If so, then you're on the right track. We have to push that ...
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Group action on a topological space and Haar integral (A question on a proof in a Palais' paper)

You have\begin{align}\int g^{-1}f(g\gamma x)\,\mathrm d\mu(g)&=\int\gamma\gamma^{-1}g^{-1}f(g\gamma x)\,\mathrm d\mu(g)\\&=\int\gamma(\color{red}{g\gamma})^{-1}f(\color{red}{g\gamma}x)\,\...
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2 votes
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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 $a$

Both of your calculations for each inclusion, the one in the question and the one in your comment are correct. To answer your question regarding "substitution", what you did was correct, it ...
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How close is $ GL_n (2) $ to being $3$ transitive?

Are there always just two orbits for the action on the space of triples, correspond to span having rank 3 versus span with rank 2 ... ? Yes. Any $3$-subset of $\mathbb{F}_2^n\setminus\{0\}$ is either ...
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1 vote

Specific questions regarding the group action $G \times \mathcal{S}(G) \to \mathcal{S}(G)$

Parts 1 and 2 are correct, but part 3 is totally wrong. I don't think you have the correct definition of transitive.
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The mean random walk on a spherical lattice excursion length

For any irreducible finite Markov chain, the expected return time to the initial state $x_0$ is $1/\pi(x_0)$, where $\pi$ is the stationary distribution. See [1] or [2], Prop. 1.19 page 13. For ...
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How to Find Orbits and Stabilizers

The formula you are using is incorrect. It is not used for the whole set, but for one element only $$\forall\ s\in S,\lvert D_6\rvert=\lvert\text{Stab}(s)\rvert\cdot\lvert\text{Orb}(s)\rvert$$ That's ...
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Power of $ 2 $ congruent to 1 mod n

Ok so just summarizing what everyone else said, the solutions are exactly the multiples of $ o_n(2) $. So the number of solutions less than $ n $ is the floor function of $ n /o_2(n) $. When 2 is ...
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In general, prove pseudo group action is proper action.

We have a pseudo-action $\star$ of $G$ on $X$. Given $g\in G,\,x\in X$, we need to find $y\in Y$ (where $Y=\{e\star x\mid x\in X\}$ is the orbit of $e$), such that $g\star x=g\star y$. Let $y=e\star x$...
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Pseudo-group action, how to prove.

For closure, you need that given $e\star x\in Y$, that $g\star(e\star x)=e\star y$ for some $y\in G$. So, let $y=g\star x$. Then $g\star (e\star x)=(ge)\star x=g\star x=(eg)\star x=e\star (g\star x)=...
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Find sizes of disjoint subsets to make transitive action by icosahedron on set $X$.

I shall expand on the hints I provided in the comments. First note that any symmetry of the icosahedron preserves distances, and so if the pair of vertices $\{u,v\}$ get transformed into $$g\star\{u,v\...
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Why $\mu_3 = \operatorname{ diag} (\zeta_3, \zeta_3)$ is the kernel of the G'-action?

Suppose $g = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in G'$ is in the kernel of the action. Then $$p((ax+by, cx+dy)) = p(g(x,y)) = (g^{-1} \cdot p)((x,y)) = p((x,y))$$ for all $p \in \...
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1 vote
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Finding stabilizer and orbit of power set of a group

The group action here $H×X\to X $ defined by $$h\cdot A=hAh^{-1}$$ Orbit: $\mathcal{O}_A=\{h\cdot A :h\in H\}=C_H(A)$ Stabilizer: $\begin{align}\operatorname{stab}_A&=\{h\in H : h\cdot A=A\}\\&...
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What condition in the definition of group action makes g.(pq) = (g.p)(g.q)?

In the linked paper, they define the group action in question immediately before Remark 2.5.1: $$(g\cdot p)(x,y)=p(g^{-1}(x,y))$$ That is, compute $g^{-1}\cdot(x,y)$ via the usual action (see below) ...
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1 vote

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

Here is an answer for the reformulated question (requiring effectiveness) but which might be considered as exploring a loophole. Further reformulations are welcome, but for now, here we go: For each $...
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1 vote

Group Operations/ Group Actions

When you have a group $G$ acting on a set $X$ (a so-called $G$-set), you get a homomorphism from $G$ into $S_X$, the symmetric group on $X$. Any group acts on itself by left (or right) multiplication. ...
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