the characteristic embedding of the Burnside ring
Let be a finite group, its subgroup
and a finite -set. By the -fixed point
subset of we understand the set
Denote by the cardinality of a set .
It is easy to see that for any -sets we have:
Denote by . Recall that any are said to be conjugate iff there exists such that . Conjugation is an equivalence relation
. Denote by the quotient set.
One can check that for any such that is conjugate to and for any finite -set we have
Thus we have a well defined ring homomorphism:
This homomorphism is known as the characteristic embedding, since it is monomorphism
(see [1] for proof).
References
- 1 T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. 766, Springer-Verlag, Berlin, 1979.