Let G have a transitive left action on a set X and set [itex] H = G_x [/itex] to be the stabilizer of any point x. Show that the map defined by [itex] f: G/H \rightarrow X [/itex] where f(gH) = gx is well defined, one to one, and onto.(adsbygoogle = window.adsbygoogle || []).push({});

i think i know how to show well defined. letting g1 H = g2 H, if i multiply on the right both sides with x, then since H is the stabilizer of x, Hx = x so i get g1 x = g2 x which is what i wanted. i am having some trouble though on showing that this function is one to one and onto most likely because i am still getting acquainted with the ideas of factor groups and cosets.

So for one to one i want to show that if g1 x = g2 x then g1 H = g2 H. what i did was [itex] x = g_1^{-1}g_2 x [/itex] which implies that [itex] g_1^{-1}g_2 \in H [/itex] and this shows [itex] g_1 H = g_2 H [/itex].

For onto i tried saying that given any gx in X, you can then find gH in G/H which maps to it. but i'm not sure if this is valid or not. when proving onto do i need to start with an arbitrary x in X? or do i start with some gx in X?

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# Showing that a map from factor group to another set bijective

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