f: G/Gs------>Os, defined by f(gGs)=gs.
Os={s'in S st s'=gs for some g in G}
Gs={g in G st gs=s}
I am supposed to show f is well defined, 1-1, and onto.
I know that to be well defined, if a=b then f(a)=f(b). In this case I think that would mean, if g1Gs=g2Gs, then g1s=g2s. So I assume that g1Gs=g2Gs is true. Does that mean that g1=g2? If so, can I then substitute a generic g in for g1 and g2?
I know 1-1 means that if f(a)=f(b), then a=b. For this case, if g1s=g2s, then g1Gs=g2Gs. So I assume g1s=g2s, and show g1Gs=g2Gs. Since g1s=g2s, g1=g2? Am I allowed to divide by s? I know I can't multiply by the reciprocal because S isn't a group, it's a set.
I know that onto means that for every element in the codomain-y, there exists and element in the domain-x with f(x)=y. So in this case, for every element in Os, the orbit of s, there must exist an element in the domain, the quotient group of Gs in G (which is the (left) cosets of Gs in G) with f(g'Gs)=g's? I really don't know what to do here?
Any guidance would be appreciated.