# Thread: normal subgroup and orbits

1. ## normal subgroup and orbits

Let the group G act on the finite set A and let $\displaystyle H \trianglelefteq G$. Let $\displaystyle O_1, O_2,\ldots,O_r$ be the distinct orbits of H on A.

$\displaystyle g \cdot a_i = a_j$, where $\displaystyle g \in G, a_i \in O_i, a_j \in O_j$
let $\displaystyle b_i = h_1 \cdot a_i \text{ for some } h_1 \in H \text{ and } a_i \neq b_i$.
now $\displaystyle g \cdot b_i = g \cdot (h_1 \cdot a_i)=(g h_1) \cdot a_i = (g h_1 g^{-1} g) \cdot a_i = (g h_1 g^{-1})g \cdot a_i = h_2 g \cdot a_i = h_2 \cdot a_j \text{ where }g h_1 g^{-1} = h_2; h_2 \in H$.
now $\displaystyle h_2 \cdot a_j = c_j \text { for some } c_j \in O_j$. also it can be shown that $\displaystyle a_j \neq c_j$ very easily.

from the above argument can i say that $\displaystyle |O_i|=|O_j|$

2. Originally Posted by abhishekkgp
Let the group G act on the finite set A and let $\displaystyle H \trianglelefteq G$. Let $\displaystyle O_1, O_2,\ldots,O_r$ be the distinct orbits of H on A.

$\displaystyle g \cdot a_i = a_j$, where $\displaystyle g \in G, a_i \in O_i, a_j \in O_j$
let $\displaystyle b_i = h_1 \cdot a_i \text{ for some } h_1 \in H \text{ and } a_i \neq b_i$.
now $\displaystyle g \cdot b_i = g \cdot (h_1 \cdot a_i)=(g h_1) \cdot a_i = (g h_1 g^{-1} g) \cdot a_i = (g h_1 g^{-1})g \cdot a_i = h_2 g \cdot a_i = h_2 \cdot a_j \text{ where }g h_1 g^{-1} = h_2; h_2 \in H$.
now $\displaystyle h_2 \cdot a_j = c_j \text { for some } c_j \in O_j$. also it can be shown that $\displaystyle a_j \neq c_j$ very easily.

from the above argument can i say that $\displaystyle |O_i|=|O_j|$

I'm not sure: why won't you directly show a bijection between $\displaystyle O_i\,,\,O_j$ ?

Tonio

3. Originally Posted by tonio
I'm not sure: why won't you directly show a bijection between $\displaystyle O_i\,,\,O_j$ ?

Tonio
Thats what i tried to do. is there another way to show a bijection between the two??

4. Originally Posted by abhishekkgp
Thats what i tried to do. is there another way to show a bijection between the two??

"Another" way? Which one has been shown? If you think that your calculations in your first post show

this then do define directly and unmistakenly the bijection, otherwise search for one...

Tonio

5. Originally Posted by tonio
"Another" way? Which one has been shown? If you think that your calculations in your first post show

this then do define directly and unmistakenly the bijection, otherwise search for one...

Tonio
In the question i forgot to mention that G is transitive on A(sorry for that).
with this in mind there exists a $\displaystyle g \in G \text{ such that } g \cdot a_i = a_j \text{ where } a_i \in O_i, a_j \in O_j$
for such a g there exists a bijection:
$\displaystyle g \colon O_i \rightarrow O_j$
$\displaystyle g \cdot m_i \mapsto m_j \text{ where } m_i \in O_i, m_j \in O_j$
to prove this consider $\displaystyle a_i, b_i \in O_i \text{ such that } a_i \neq b_i$
now $\displaystyle b_i = h_1 \cdot a_i \text{ for some } h_1 \in H$
from my first post $\displaystyle g \cdot b_i = (g h_1 g^{-1})g \cdot a_i = h_2 g \cdot a_i = h_2 \cdot a_j$
to show that the mapping is injective i must show that $\displaystyle g \cdot a_i \neq g \cdot b_i, \text{ that is, } a_j \neq h_2 \cdot a_j$
i will use contradiction to do so. Assume $\displaystyle a_j=h_2 \cdot a_j \Rightarrow a_j=(g h_1 g^{-1}) \cdot a_j \Rightarrow a_j= (g h_1) \cdot a_i \Rightarrow a_j=g \cdot b_i \Rightarrow g^{-1} \cdot a_j = b_i \Rightarrow a_i =b_i$ which is a contradiction. so the map is injective.

has the injection been shown correctly? If this is correct then i can attempt to show the surjection too. Please comment.

6. Originally Posted by abhishekkgp
In the question i forgot to mention that G is transitive on A(sorry for that).
with this in mind there exists a $\displaystyle g \in G \text{ such that } g \cdot a_i = a_j \text{ where } a_i \in O_i, a_j \in O_j$
for such a g there exists a bijection:
$\displaystyle g \colon O_i \rightarrow O_j$
$\displaystyle g \cdot m_i \mapsto m_j \text{ where } m_i \in O_i, m_j \in O_j$

Uuh?? You define a function between two sets naming it the same as an element in G? And besides this,

how is the function defined? Because we know that $\displaystyle g\cdot a_i=a_j$ , but what happens with some

$\displaystyle a_i\neq x\in O_i??$ How's defined the map you're talking about on x? Can you be sure that $\displaystyle g\cdot x\in O_j$

to begin with at all? So, how do you define the map $\displaystyle g$ ?

to prove this consider $\displaystyle a_i, b_i \in O_i \text{ such that } a_i \neq b_i$
now $\displaystyle b_i = h_1 \cdot a_i \text{ for some } h_1 \in H$

from my first post $\displaystyle g \cdot b_i = (g h_1 g^{-1})g \cdot a_i = h_2 g \cdot a_i = h_2 \cdot a_j$
to show that the mapping is injective i must show that $\displaystyle g \cdot a_i \neq g \cdot b_i, \text{ that is, } a_j \neq h_2 \cdot a_j$
i will use contradiction to do so. Assume $\displaystyle a_j=h_2 \cdot a_j \Rightarrow a_j=(g h_1 g^{-1}) \cdot a_j \Rightarrow a_j= (g h_1) \cdot a_i \Rightarrow a_j=g \cdot b_i \Rightarrow g^{-1} \cdot a_j = b_i \Rightarrow a_i =b_i$ which is a contradiction. so the map is injective.

has the injection been shown correctly? If this is correct then i can attempt to show the surjection too. Please comment.
.

This looks fine, if we address the above questions I wrote.

About surjectivity; Since $\displaystyle |O_i|=|O_j|<\infty$ , a function between them is

injective iff it is surjective.

Tonio

7. I have committed a mistake in defining the bijection, what i should have written is the following:
$\displaystyle g \colon O_i \rightarrow O_j$
$\displaystyle x_i \mapsto g \cdot x_i$

you have asked whether it can be said for sure that $\displaystyle g \cdot x \in O_j \text{ when } x \in O_i, x \neq a_i$.
I show that it is sure to hold. proof: since $\displaystyle x \in O_i \text{ there exists } h \in H \text{ such that } h \cdot a_i = x$
now $\displaystyle g \cdot x = g \cdot ( h \cdot a_i)$
$\displaystyle \Rightarrow g \cdot x = (g h g^{-1})g \cdot a_i$
but we know that $\displaystyle g \cdot a_i =a_j \text{ and } g h g^{-1} \in H$
$\displaystyle \Rightarrow g \cdot x = h^\prime \cdot a_j$
now since $\displaystyle h^\prime \cdot a_j \in O_j \text{ we have } g \cdot x \in O_j$

Thank you for your help. is the definition of the bijection alright now? it was a blunder on my part