Let the group G act on the finite set A and let . Let be the distinct orbits of H on A.
now read the following:
now . also it can be shown that very easily.
from the above argument can i say that
with this in mind there exists a
for such a g there exists a bijection:
to prove this consider
from my first post
to show that the mapping is injective i must show that
i will use contradiction to do so. Assume 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.
I have committed a mistake in defining the bijection, what i should have written is the following:
you have asked whether it can be said for sure that .
I show that it is sure to hold. proof: since
but we know that
Thank you for your help. is the definition of the bijection alright now? it was a blunder on my part