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6.2: Órbitas y Estabilizadores

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    En esta sección, examinaremos órbitas y estabilizadores, lo que nos permitirá relacionar acciones grupales con nuestro estudio previo de coconjuntos y cocientes.

    Definición 6.1.0: La órbita

    Let\(S\) be a \(G\)-set, and \(s\in S\). The orbit of \(s\) is the set \(G\cdot s = \{g\cdot s \mid g\in G\}\), the full set of objects that \(s\) is sent to under the action of \(G\).

    Hay algunas preguntas que surgen al encontrarse con una nueva acción grupal. Lo más importante es 'Dados dos elementos\(s\) and \(t\) from the set \(S\), is there a group element such that \(g\cdot s=t\)?' In other words, can I use the group to get from any element of the set to any other? In the case of the action of \(S_n\) on a coin, the answer is yes. But in the case of \(S_4\) acting on the deck of cards, the answer is no. In fact, this is just a question about orbits. If there is only one orbit, then I can always find a group element to move from any object to any other object. This case has a special name.

    Definición 6.1.1: Acción Transitiva de Grupo

    Una acción de grupo es transitiva si\(G\cdot s = S\). In other words, for any \(s, t\in S\), there exists \(g\in G\) such that \(g\cdot s=t\). Equivalently, \(S\) contains a single orbit.

    Igualmente importante es el estabilizador de un elemento, el subconjunto de\(G\) which leaves a given element \(s\) alone.

    Definición 6.1.2: El Estabilizador

    El estabilizador de\(s\) is the set \(G_s = \{g\in G \mid g\cdot s=s \}\), the set of elements of \(G\) which leave \(s\) unchanged under the action.

    Por ejemplo, el estabilizador de la moneda con cabezas (o colas) hacia arriba es\(A_n\), the set of permutations with positive sign. In our example with \(S_4\) acting on the small deck of eight cards, consider the card \(4D\). The stabilizer of \(4D\) is the set of permutations \(\sigma\) with \(\sigma(4)=4\); there are six such permutations.

    En ambos ejemplos, el estabilizador era un subgrupo; ¡esto es un hecho general!

    Proposición 6.1.3

    El estabilizador\(G_s\) of any element \(s \in S\) is a subgroup of \(G\).

    Prueba 6.1.4

    Let\(g, h \in G_s\). Then \(gh\cdot s = g\cdot (h\cdot s) = g\cdot s=s\). Thus, \(gh\in G_s\). If \(g\in G_s\), then so is \(g^{-1}\): By definition of a group action, \(1\in G_s\), so:
    \(s=1\cdot s = g^{-1}g\cdot s = g^{-1}s\).

    Por lo tanto,\(G_s\) is a subgroup.


    Morfismos de acción grupal

    ¡Y ahora algunos ejemplos algebraicos!

    1. Let\(G\) be any group and \(S=G\). The left regular action of \(G\) on itself is given by left multiplication: \(g\cdot h = gh\). The first condition for a group action holds by associativity of the group, and the second condition follows from the definition of the identity element. (There is also a right regular action, where \(g\cdot h = hg\); the action is 'on the right'.) The Cayley graph of the left regular action is the same as the usual Cayley graph of the group!
    2. Let\(H\) be a subgroup of \(G\), and let \(S\) be the set of cosets \(G/\mathord H\). The coset action is given by \(g\cdot (xH) = (gx)H\).

    This page titled 6.2: Órbitas y Estabilizadores is shared under a not declared license and was authored, remixed, and/or curated by Tom Denton.