We next need to show that . We know that and, using the first criterion for an action, we get the required result.No.2). Show that there is a single orbit, , undder this action.
By the Orbit-Stabiliser theorem, I know that .
4). Show that the kernel of the action is a normal subgroup of that is contained in .
I know that .
The elements that satisfy this are going to be all . Therefore, .
(^Is this right?)
This then gives the required result: .
Let M be the set of all left cosets of H in G.
You need to show that if xH and yH are two arbitrary left cosets in M, then you can find an element such that . In this case, choose g as . This means, G acts transitively on M.
Let M be the set of all left cosets of H in G. You need to check the induced homomorphism defined by , where and , where A(M) is the group of permutations of M. If g is in the kernal gxH=xH for all x in G, esp, geH=eH, then g should be in H.I need to show that when .
by substituting in .
I know that is the kernel of the action of on .
Rewriting this as gives that the only element common to every set is when .
The set is indeed a normal subgroup, and must be in since every subgroup of a group shares the same identity element.
The induced homomorphism defined by the previous question can be used here. The kernel of h is contained in H, actually it has the form and it is the largest normal subgroup of G contained in H.
we have left cosets.
THerefore choose to be the subgroup of all left cosets of . This gives a quotient group .
Define a homomorphism by .
I have a proposition in my notes stating that and that is a homomorphism.
From here I get stuck. I've been trying to apply the first isomorphism theorem, but I have no idea how to put it into action.
Anyway, any help would be appreciated.
It follows that G/N is isomorphic to the subgroup of A(M), where A(M) is the group of all permutations of M isomorphic to .
Edit: h is not necessarily a surjective homomorphism. Thus you can't use the first isomorphism theorem here. Rather, you need to show that G/N can be embedded into A(M).