Group Action, Double costs

Let H,K subgroups of G for each x in G define the double cost HK of x

a) Prove that HxK is the union of the left costs where is the orbit containing xK of H acting by left Multiplication on the set of left costs of K

My work

I want to prove that it is clear that

so

let hx in G so hxK in the union

d) Prove that

my question is about d) I put a) because I think it will help in solving d)

I was thinking about the order of orbit equal to the index group of the stabilizer

so we will have

which is not like what i want to prove

THanks

Re: Group Action, Double costs

Re: Group Action, Double costs

Quote:

Originally Posted by

**Drexel28** I think you are going about the last part wrong. You know that

should be equal to

where

is orbit of

under the left

-action on

. But, each

is equal to

and so

. But, using the orbit stabilizer theorem, as you indicated, you can show that

and so

.

For more information you can see my blog post

here.

EDIT: I noticed that I did the "opposite side" than you wanted, but by symmetry they're the same.

Thanks very much for the direction I did not noticed that but there still a small problem the question show that

not

you can check the book David Dammit same source which you gave in your Blog

chapter 4

is it a typo ?

nice blog I like it (Nerd)

Re: Group Action, Double costs

Quote:

Originally Posted by

**Amer** Thanks very much for the direction I did not noticed that but there still a small problem the question show that

not

you can check the book David Dammit same source which you gave in your Blog

chapter 4

is it a typo ?

nice blog I like it (Nerd)

What I wrote is correct, namely . Of course, since (by the obvious bijection) we also have that .

And, thanks about the blog!

Re: Group Action, Double costs

when i said typo i mean in the book I made an example

let , ,

H acting on xK by left additive

the orbit of H acting on the left cost will be

but if it is which is not correct since the order should equal to the G order

so there is a typo in book

am I right ?

Re: Group Action, Double costs

Quote:

Originally Posted by

**Amer** when i said typo i mean in the book I made an example

let

,

,

H acting on xK by left additive

the orbit of H acting on the left cost will be

but if it is

which is not correct since the order should equal to the G order

so there is a typo in book

am I right ?

Yes, you are correct. There is a typo. It is not but .

Re: Group Action, Double costs

Quote:

Originally Posted by

**Drexel28** Yes, you are correct. There is a typo. It is not

but

.

Thanks for the fast respond I appreciate that