Thread: G-set

Hi


1)Given G, a group, and X a G-set What relationship exists between and if x and y belong to the same orbit?


2)

a)Let G be a group and X a G-set.Prove that there is a homomorphism
determined by the action.

b)Now, given a homomorphism
, show that X is a G-set

I hope some help. thanks

1) if x,y belong to the same orbit, then there is some g in G with g.x = y. thus g^-1.y = g^-1.(g.x) = (g^-1g).x = e.x = x

now suppose that a stabilizes x (that is, a is in G_x), that is: a.x = x.

then for the g for which g.x = y, we have:

(gag^-1).y = g.(a.(g^-1.y)) = g.(a.x) = g.x = y, so gag^-1 stabilizes y.

on the other hand, suppose that b.y = y. note that b = ebe^-1 = g(g^-1bg)g^-1.

now (g^-1bg).x = g^-1.(b.(g.x)) = g^-1.(b.y) = g^-1.y = x.

this means that G_y = gG_xg^-1, G_x and G_y are conjugate subgroups.

2) suppose X is a G-set.

then we have a mapping G x X → X given by (g,x) → g.x with (for all g,h in G, and x in X):

g.(h.x) = (gh).x
e.x = x

define φ:G → Sym(X) by: φ(g)(x) = g.x (that is φ(g) is the function that takes x to g.x, so φ(g):X → X).

we need to show that φ(g) is a bijection on X, and that φ(gh) = φ(g)○φ(h).

note that that (φ(g)○φ(g^-1))(x) = φ(g)(φ(g^-1)(x)) = φ(g)(g^-1.x) = g.(g^-1.x) = (gg^-1).x = e.x = x, and:

(φ(g^-1)○φ(g))(x) = φ(g^-1)(φ(g)(x)) = φ(g^-1)(g.x) = g^-1.(g.x) = (g^-1.g)(x) = e.x = x.

thus φ(g^-1) is a two-sided inverse for φ(g), so φ(g) is a bijection.

now φ(gh)(x) = gh.x = g.(h.x) = φ(g)[φ(h)(x)] = (φ(g)○φ(h))(x), for all x in X, thus φ(gh) = φ(g)○φ(h), φ is a homomorphism.

on the other hand, suppose we just have a homomorphism φ of G into Sym(X).

we can make X into a G-set as follows:

define g.x = φ(g)(x). all we have to do is verify this satisfies the rules for a G-set.

note that since φ is a homomorphism, it has to map the identity of G, to the identity map of X (since that is the identity of the group Sym(X)). this means that:

φ(e)(x) = x, for all x in X, so e.x = x.

also, since φ is a homomorphism, φ(gh) = φ(g)○φ(h), which means that:

(gh).x = φ(gh)(x) = (φ(g)○φ(h))(x) = φ(g)(φ(h)(x)) = g.(φ(h)(x)) = g.(h.x), so we have a G-set.