Prove that G acts transitively on normal subgroup orbits on nonempty finite set A

The Question is let acts transitively on a nonempty finite set A

let H be normal subgroup of G,Let Orbits of H on A be

prove that G acts transitively on

My work I want to prove that for any two oribts there exist such that

I will call elements with "a" and with "b"

pick

there exist g in G such that since G acts transitively on A

let

want to show that is an element of

but H is normal subgroup so b_s in O_i

b_s is arbitrary so for any b in O_j g.b is an element of O_i

ends

is it correct ?? is there any shorter way ? any ideas

Thanks

Re: Prove that G acts transitively on normal subgroup orbits on nonempty finite set A

is it already given that a normal subgroup H of G induces a well-defined action on the orbits of H? (it's true, but what i'm asking is: have you already proven this earlier?)

aside from that, your proof of the transitivity of this action of G on the orbits of H looks fine to me.

Re: Prove that G acts transitively on normal subgroup orbits on nonempty finite set A

Quote:

Originally Posted by

**Deveno** is it already given that a normal subgroup H of G induces a well-defined action on the orbits of H? (it's true, but what i'm asking is: have you already proven this earlier?)

aside from that, your proof of the transitivity of this action of G on the orbits of H looks fine to me.

you are wright it is not the whole question i proved that before, any orbit of H say O, gO still orbit of H

let a,b in orbit O, we have a = h.b for some h in H

if I can show that g.a , g.b is in the same orbit I am done

so g.a , g.b in same orbit in H

is it Okay ??

Re: Prove that G acts transitively on normal subgroup orbits on nonempty finite set A

that's right, and it shows why H has to be normal for this to work.

Re: Prove that G acts transitively on normal subgroup orbits on nonempty finite set A

in the same question i asked to prove that all orbits has the same cardinality

My work,I will show that the stabalizer of each representative element of the orbits has the same cardinality, let

orbits of H on A let a,b be the element that present each orbit

Stab of a in H so

and which end the proof

I did not post another thread because it is one question

is it correct ?

Thanks