### 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 *_{G}X 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 }.
*

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001