# 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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### Quotient of Hausdorff space by free discrete group action is Hausdorff

Let $X$ be a Hausdorff topological space and $G$ a group of homeomorphisms from $X$ to $X$. Suppose that $G$ acts on $X$ discretely, that is, for any $a,b$ in $X$, there exists a neighbourhood $U_x$ ...
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### Isometry group of $\mathbb{Z}$

Consider the group of integers $\mathbb{Z}$ equipped with the discrete metric $$d(m, n) = \begin{cases} 1, \quad m \neq n\\ 0 , \quad m = n \end{cases}$$ In particular, $\mathbb{Z}$ is a metric Lie ...
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### elaboration of group action definition.

My professor wrote at the beginning of speaking about group actions this: In general, Aut$(X) \subset$ Sym$(X)$ acts on $X$. If $G \subset Aut(X)$ is a subgroup, we say that "G acts on $X$ by ...
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### Trivial group action

On page 43 of Dummit & Foote's abstract algebra: Let $G$ be a group and $A$ a nonempty set. Let $ga = a$, for all $g \in G$, $a \in A$. This action is called the trivial action and $G$ is said to ...
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### About groups act faithfully on a set

Suppose group $G$ act faithfully on a set $X$ of $5$ elements, and there are $2$ orbits, of order $2$ and $3$ respectively. Then what should the group $G$ be like? Note: A group $G$ acts faithfully on ...
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### Classify all maps $f: (\mathbb{R}^d)^n \to \mathbb{R}^d$ such that $f(R x_1,\dotsc, R x_n) = R f(x_1,\dotsc, x_n)$ for all $R \in O(d)$

Let $O(d)$ denote the orthogonal group for $\mathbb{R}^d$ with the standard Euclidean inner-product. A nautral question is to find all possible functions $f: \mathbb{R}^d \to \mathbb{R}$ that are ...
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### Semigroup action generalized (relations instead of functions)

If with every element of a (semi)group is associated a function, it is basically called a (semi)group action. What if with each element of a semigroup is associated a relation? That is, formally, what ...
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### Confusion on how groups acts on objects

im trying to learn group theory to understand particle physics and im currently reading: "A Simple Introduction to Particle Physics ,Part I - Foundations and the Standard Model". In the ...
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### Sharply $k$-transitive actions on spheres

A nice fact from complex analysis is that the mobius group acts sharply 3-transitively on the Riemann sphere. I am wondering if other sharply k-transitive (continuous) actions are known on any $S^n$, ...
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### Count the number of orbits under the action of $S_4$ on $\mathcal P (X),$ where $X = \{1,2,3,4 \}.$

Let the symmetric group $S_4$ act on $X = \{1,2,3,4 \}.$ This gives an action on the power set $\mathcal P(X).$ Count the number of orbits for the action of $S_4$ on $\mathcal P(X).$ My attempt $:$ ...
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