The following is a very important application of actions on k-subsets:
The regular representation of G
yields, in accordance with formula, the G-sets
[G choose k],
for 1 <= k <= | G | . If G is finite and p a prime
dividing | G | , say | G | = pr ·q, r ³1,
q=pst, where p does not divide t, then we can put k:=pr and
consider the particular G-set [G choose pr],
as H. Wielandt did in his famous
proof of Sylow's Theorem
in order to show that G possesses
subgroups of order pr. His argument runs as follows:
ps is the exact power of p dividing the order of
[G choose pr]. This is
|[G choose pr] | =
(prq)/(pr) ·(prq-1)/(1) ...(prq-(pr-1))/(pr-1),
as each power of p contained in the denominator cancels.
Thus pr-subsets M exist, the orbit length
of which is not divisible by ps+1.
We consider such an M and show that its stabilizer GM is of
order pr by proving that pr is both an upper and lower bound:
For each m ÎM and g ÎGM we have that gm ÎM, hence
| GM | <= | M | = pr.
On the other hand, the fact that ps+1 does not divide the orbit length
| G(M) | =
| G | / | GM | yields
| GM | >= pr.
This proves the first item of
Assume G to be a finite group
and p to be a prime divisor of its order. Then
The proof of the second and third item follows from a consideration of double
cosets. Assume a p-subgroup U of G and a Sylow p-subgroup S. Then
we derive from the corollary that
The subgroups S £G of the maximal p-power order are called
the Sylow p-subgroups
of G. They have the following properties:
- G contains
subgroups of order pr, for each power pr dividing its
order | G | .
- Each p-subgroup U of G is contained
in a suitable Sylow p-subgroup S.
- Any two Sylow
p-subgroups of G are conjugate subgroups.
( | G | )/( | S | )= åg Î D( | U | )/( | U ÇgSg-1 | ),
where D denotes a transversal of U \G/S.
If all the intersections in the denominator on the right hand side were proper subgroups
of U, then the right hand side were divisible by p, which contradicts the left hand
side. Hence there must exist a g ÎD, such that U £gSg-1. Since
gSg-1 is a Sylow p-subgroup, too, U is contained in a Sylow subgroup, which
proves the second item.
The third item follows by taking for U a Sylow p-subgroup S':
( | G | )/( | S | )= åg Î D'( | S' | )/( | S' ÇgSg-1 | )
shows that S'=gSg-1, for a suitable g Î D',
where D' denotes a transversal of S' \G/S.
This example shows clearly that the consideration of suitable group
actions can be very helpful, at least in group theory. Applications
to other fields of mathematics will follow soon.
last changed: January 19, 2005