the double coset counting formula is a relation inter double cosets , where and subgroups in . This is:
and
One is to be bounded to the study of the natural map . And it uses the second abstraction lemma.
The formula allows you to see the kinds of subgroups of arbitrary versus a of , for the set of the – Sylow subgroups.
Or, you can see that through the action via you can get:
- which comply the equi-partition
- , so , for some
then you can deduce:
Now, let us use those ideas to prove the next statement:
Let be a finite group, with cardinal , where each are primes with and positive integers.
Let be a subgroup of of index .
Then, is normal.
Proof:
By employing in the double coset partition, one get the decomposition:
So by the double coset counting formula you arrive to:
i.e.
From this, we get .
But as well so
, i.e.
divides
Then . So for each .
This implies and so for all the posible , hence, is normal.
QED.
take a look http://projecteuclid.org/download/pdf_1/euclid.bams/1183503688
Reblogged this on janmarqz.
Cf. http://drexel28.wordpress.com/2011/09/16/a-clever-proof-of-a-common-fact/
Cf. http://drexel28.wordpress.com/2011/05/02/double-cosets/