Let be a subgroup of a group and let be the set of left cosets of in .I know that . is a group closed under the binary operation ".". Therefore if , then . This is true so the first criterion for an action is satisfied.1). Show that defines an action of on .
We next need to show that . We know that and, using the first criterion for an action, we get the required result.
By the Orbit-Stabiliser theorem, I know that .2). Show that there is a single orbit, , undder this action.
I know that .
The elements that satisfy this are going to be all . Therefore, .
(^Is this right?)
This then gives the required result: .
I need to show that when .3). Show that the stabiliser of the coset is .
by substituting in .
I know that is the kernel of the action of on .4). Show that the kernel of the action is a normal subgroup of that is contained in .
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.
we have left cosets.Show that if with then has a quotient group which is isomorphic to a subgroup of the symmmetric group with .
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.