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.