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 | = å_{g Î D} | AgB | = å_{g 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 | C_G(a) | = | C_G(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} ( | C^G(a) ÇC^G(b) | )/( | C^G(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 | ) å_{g ÎC}( | C^G(g) ´C^G(g) ÇU | )/( | C^G(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 | ) å_{g ÎC}( | C^G(g) ÇA | | C^G(g) ÇB | )/( | C^G(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/G_x :U(gx) -> UgG_x.

In particular, a transversal D of the set of double cosets U \G/G_x yields the following transversal of the set of orbits of U:

T(U \\X):= {gx | g Î D }.

Double Cosets