Index/Isomorphism/Quotient Group answer check

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Let

be a subgroup of a group

and let

be the set of left cosets

of

in

.

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1). Show that

defines an action of

on

.

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.

We next need to show that . We know that and, using the first criterion for an action, we get the required result.

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2). Show that there is a single orbit,

, undder this action.

By the Orbit-Stabiliser theorem, I know that .

I know that .

The elements that satisfy this are going to be all . Therefore, .

(^Is this right?)

This then gives the required result: .

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3). Show that the stabiliser of the coset

is

.

I need to show that when .

by substituting in .

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4). Show that the kernel of the action is a normal subgroup of

that is contained 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.

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Show that if

with

then

has a quotient group

which is isomorphic to a subgroup

of the symmmetric group

with

.

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.