Group Action, Double costs

**Amer** · November 19th 2011, 09:03 AM

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

$HxK=\{hxk\mid h\in H ,k\in K\}$

a) Prove that HxK is the union of the left costs $x_1K,x_2K,...,x_nK$ where $\{x_1K,x_2K,...,x_nK\}$ 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 $HxK = \bigcup_i^n x_iK$ it is clear that
$1xK = xK$ so $\bigcup_i^n x_iK \subseteq HxK$

let $hxK \in HxK$ hx in G so hxK in the union

d) Prove that $\mid HxK \mid = \mid H \mid .\mid H : H\bigcap xKx^{-1}\mid$

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

$\mid HxK\mid = \mid \bigcup x_iK\mid = \mid O \mid = \mid H: Stab_H(O)\mid$

$Stab_H(O)= \{ h \in H \mid hxK = xK \}$

$Stab_H(O) = \{h \in H \mid x^{-1}hx \in K \Rightarrow h\in xKx^{-1}\}$

$Stab_H(O) = H \bigcap xKx^{-1}$
so we will have

$\mid HxK\mid = \mid H: H \bigcap xKx^{-1}\mid$ which is not like what i want to prove

THanks

**Drexel28** · November 19th 2011, 01:04 PM

Originally Posted by Amer

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

$HxK=\{hxk\mid h\in H ,k\in K\}$

a) Prove that HxK is the union of the left costs $x_1K,x_2K,...,x_nK$ where $\{x_1K,x_2K,...,x_nK\}$ 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 $HxK = \bigcup_i^n x_iK$ it is clear that
$1xK = xK$ so $\bigcup_i^n x_iK \subseteq HxK$

let $hxK \in HxK$ hx in G so hxK in the union

d) Prove that $\mid HxK \mid = \mid H \mid .\mid H : H\bigcap xKx^{-1}\mid$

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

$\mid HxK\mid = \mid \bigcup x_iK\mid = \mid O \mid = \mid H: Stab_H(O)\mid$

$Stab_H(O)= \{ h \in H \mid hxK = xK \}$

$Stab_H(O) = \{h \in H \mid x^{-1}hx \in K \Rightarrow h\in xKx^{-1}\}$

$Stab_H(O) = H \bigcap xKx^{-1}$
so we will have

$\mid HxK\mid = \mid H: H \bigcap xKx^{-1}\mid$ which is not like what i want to prove

THanks

I think you are going about the last part wrong. You know that $HxK$ should be equal to $\displaystyle \bigsqcup_{gK\in\mathcal{O}_{xK}}|gK|$ where $\mathcal{O}_{xK}$ is orbit of $xK$ under the left $H$ -action on $G/K$ . But, each $|gK|$ is equal to $|K|$ and so $|HxK|=|K||\mathcal{O}_{xK}|$ . But, using the orbit stabilizer theorem, as you indicated, you can show that $|\mathcal{O}_{xK}|=[H:H\cap xKx^{-1}]$ and so $|HxK|=|K|[H:H\cap xKx^{-1}]$ .

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.

**Amer** · November 19th 2011, 06:43 PM

Originally Posted by Drexel28

I think you are going about the last part wrong. You know that $HxK$ should be equal to $\displaystyle \bigsqcup_{gK\in\mathcal{O}_{xK}}|gK|$ where $\mathcal{O}_{xK}$ is orbit of $xK$ under the left $H$ -action on $G/K$ . But, each $|gK|$ is equal to $|K|$ and so $|HxK|=|K||\mathcal{O}_{xK}|$ . But, using the orbit stabilizer theorem, as you indicated, you can show that $|\mathcal{O}_{xK}|=[H:H\cap xKx^{-1}]$ and so $|HxK|=|K|[H:H\cap xKx^{-1}]$ .

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

$\mid HxK \mid = \mid H \mid . \mid H : H \bigcap xKx^{-1} \mid$
not
$\mid HxK \mid = \mid K \mid . \mid H : H \bigcap xKx^{-1} \mid$

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

**Drexel28** · November 19th 2011, 07:30 PM

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

$\mid HxK \mid = \mid H \mid . \mid H : H \bigcap xKx^{-1} \mid$
not
$\mid HxK \mid = \mid K \mid . \mid H : H \bigcap xKx^{-1} \mid$

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

What I wrote is correct, namely $|HxK|=|K|[H:H\cap xKx^{-1}]$ . Of course, since $|HxK|=|KxH|$ (by the obvious bijection) we also have that $|HxK|=|H|[K:K\cap xHx^{-1}]$ .

And, thanks about the blog!

**Amer** · November 19th 2011, 08:00 PM

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

let $G=Z_{24}$ , $H = <3> = \{0,3,6,9,...,21\}$ , $K = <4> = \{0,4,8,...,20\}$

$\mid H\mid = 8 , \mid K \mid = 6$

H acting on xK by left additive
$G/K = \{K, 1+K,2+K,3+k\}$

the orbit of H acting on the left cost will be $\{K, 1+K,2+K,3+k\}$

$\mid HxK \mid = \mid K \mid . \mid H:Stab_H(xK)\mid = 6.4 = 24$

but if it is $\mid HxK \mid = \mid H \mid . \mid H :Stab_H(xK) \mid = 8.4 = 32 ??$ which is not correct since the order should equal to the G order
so there is a typo in book
am I right ?

**Drexel28** · November 19th 2011, 08:02 PM

Originally Posted by Amer

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

let $G=Z_{24}$ , $H = <3> = \{0,3,6,9,...,21\}$ , $K = <4> = \{0,4,8,...,20\}$

$\mid H\mid = 8 , \mid K \mid = 6$

H acting on xK by left additive
$G/K = \{K, 1+K,2+K,3+k\}$

the orbit of H acting on the left cost will be $\{K, 1+K,2+K,3+k\}$

$\mid HxK \mid = \mid K \mid . \mid H:Stab_H(xK)\mid = 6.4 = 24$

but if it is $\mid HxK \mid = \mid H \mid . \mid H :Stab_H(xK) \mid = 8.4 = 32 ??$ 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 $|HxK|=|H|[H:H\cap xKx^{-1}]$ but $[HxK]=|K|[H:H\cap xKx^{-1}$ .

**Amer** · November 19th 2011, 08:03 PM

Originally Posted by Drexel28

Yes, you are correct. There is a typo. It is not $|HxK|=|H|[H:H\cap xKx^{-1}]$ but $[HxK]=|K|[H:H\cap xKx^{-1}$ .

Thanks for the fast respond I appreciate that

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