Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

When describing a group action, it should be clear whether the action is a left action or a right action. A right action is a function,

$\cdot : X\times G\rightarrow X$

such that $x\cdot 1=x$ for all $x\in X$ and such that $(x\cdot g)\cdot h=x\cdot (gh)$. These conditions mean the action of the group makes sense; that the action is compatible with the group.

A left action is defined analogously.

If $G$ acts on $X$ then there exists a homomorphism of groups $G\rightarrow \operatorname{Aut}(X)$. This is of interest when $X$ is a group too, and allows us to construct semidirect products of groups.

For more details, see Wikipedia.

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how to convert right group action to left group action?

In Wikipedia it says one can convert right group action to left group action, because of the formula $(gh)^{−1} = h^{−1}g^{−1}$. Can you explain how this works?
den
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Confused over group action

Say I want to put action of $S_4$ on $V^{\otimes 4}$. If I want to act on the left, why can't I say $$\sigma(v_1 \otimes v_2 \otimes v_3 \otimes v_4) = v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(4)}?$$ Everything seems to work out right e.g.…
user23086
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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 $\ker \phi = \{g | gx =x \}$ (so ker is a stabilizer).So in…
Lemon
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How to prove this action is discontinuous?

Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if $G_x :=$ the stabilizer of $x$, is finite. $x$ has an open neighbourhood $U$ such that $gU \cap U =…
R_D
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Group action on $\mathbb R^2$: are my thoughts correct?

Let $G=\mathbb Z / n \mathbb Z$ for $n > 2$ and let $G$ act on $\mathbb R^2$ linearly and effectively. Let $T_g (v)$ denote the element $gv$ where $v \in \mathbb R^2$. Assume that $\det T_g > 0$ and that $\langle T_g v, T_g v'\rangle = \langle…
user174981
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Action of the Unitary Group

I am working on the space $V_k (\mathbb{C}^n) = \left\lbrace (v_1, \cdots , v_k ) \in (\mathbb{C}^n)^k | \langle v_i, v_j \rangle = \delta_{ij} \right\rbrace $. I define the continuous action of $U(n)$ on $V_k (\mathbb{C}^n)$ by $ U \cdot (v_1,…
Mia
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Quotient by the action of positive reals

Today in my complex analysis class Riemann sphere was defined, and of course the construction caused questions, such as "why don't we distinguish between all the various infinities?" and "Would it work for reals, so as to distinguish between…
user22835
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How do I show that these are bijective actions?

I have the following problem: We fix a group G and a set X. I have just shown the following two statements. Let $a:G\times X\rightarrow X$ be a left action of G on X. Show that the map $b:X\times G\rightarrow X, b(x,g)=a(g^{-1},x)=g^{-1}x$ defines…
user123234
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How to find the inverse of group actions

Let $G$ and $H$ be two groups and let $$\alpha : H~\times~G \rightarrow H$$ be a right action of $G$ on $H$ so that $$\alpha_{g}(h) = h\prime$$ where $g\in G$, $h, h\prime\in H.$ I want to write the inverse of $h\prime$ in terms of $g$ and $h$.…
Dziedzorm
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Torus action on compact manifolds

Let T = $S^1 \times ...\times S^1 $be a torus acting on a compact manifold M. Let $m\in M $ and $t_m = \lbrace X \in Lie(T), X.m = 0 \rbrace$. Why is the set $ t=\lbrace t_m , m\in M \rbrace$ finite ? And does this result holds in a larger context,…
Maria
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Alternating groups of order n, group action on set of m elements

Let $A_n$, $n \geq 5$ act transitively on a set with $m>1$ elements, then I need to show that $m \geq n$. I have been thinking of $A_n$ being simple, and somehow figure out something out of this. Perhaps, I should choose a group action $A_n$ x…
user511110
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Group $ GL(V) $ acts naturally on $ \mathscr{F} $, how to get its orbits?

Could you please explain to me the meaning of the marking? Thanks in advance.
Ryze
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Does an effective group action on a compact manifold has $G$ as an orbit?

Recall that a compact manifold $M$ with a $G$-action, where $G$ is a compact Lie group, is such that $M$ contains an open, dense and convex subset where the points have the smaller possible isotropy group. Assuming the action is effective, does $M$…
L.F. Cavenaghi
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Consider the natural action of $G$ on the set $\{1,2,3,4\}$. Write down the orbits of the action and find the stabilizer of each point

Let $G\leq S_4$ and consider the natural action of $G$ on the set $\{1,2,3,4\}$. For each of the following choices of $G$, write down the orbits of the action and find the stabilizer of each point. $G=\big <(123)\big…
Leyla Alkan
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action of general linear group on the set of lines through origin

Let $A $ denotes the set of all lines through the origin.Let $G=GL_{n}(\mathbb{R})$ be the general linear group of $ 2 \times 2 $ matrices and $H$ be the subgroup of $G$ containing all lower triangular matrices. Then find a line $l$ in $A$ whose…
mmath
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