Double Cosets

### Double Cosets

Corollary: If G denotes a finite group with subgroups A and B, then
| AgB | =( | A | | B | )/( | A ÇgBg-1 | ),
and if D denotes a transversal of the set A \G/B of (A,B)-double cosets, then
| G | = åg Î D | AgB | = åg Î D( | A | | B | )/( | A ÇgBg-1 | ).

The Cauchy-Frobenius Lemma yields the number of bilateral classes. In order to evaluate it we have to calculate the number | G(a,b) | of fixed points of (a,b) ÎU, which is

| {g | a=gbg-1 } |

which equals | CG(a) | = | CG(b) | if a, b are conjugates, and which is 0 otherwise.

Thus, by the Cauchy-Frobenius Lemma, we obtain

| U \\G | =( | G | )/( | U | ) å(a,b) ÎU ( | CG(a) ÇCG(b) | )/( | CG(a) | 2).
Corollary: If U denotes a subgroup of G ´G, G being a finite group, then the number of bilateral classes of G with respect to U is
( | G | )/( | U | ) åg ÎC( | CG(g) ´CG(g) ÇU | )/( | CG(g) | ),
if C denotes a transversal of the conjugacy classes of elements in G. In particular, the set
A \G/B:= {AgB | g ÎG }=(A ´B) \\G
of (A,B)-double cosets has the order
| A \G/B | =( | G | )/( | A | | B | ) åg ÎC( | CG(g) ÇA | | CG(g) ÇB | )/( | CG(g) | ) .

The main reason for the fact that double cosets show up nearly everywhere in the applications of group actions is the following one (which we immediately obtain from lemma):

Corollary: If GX is transitive and U denotes a subgroup of G, then, for each x ÎX we have the natural bijection
j:U \\X -> U \G/Gx :U(gx) -> UgGx.
In particular, a transversal D of the set of double cosets U \G/Gx yields the following transversal of the set of orbits of U:
T(U \\X):= {gx | g Î D }.

 harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/ UNI-Graz Institut für Mathematik UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

 Double Cosets