Let be a finite group. Recall that by -set we understand a pair , where is a set and is a group action of on . For short notation the pair notation will be omitted and -sets will be simply denoted by capital letters.
Recall that for each subgroup we have canonical -set where group action is defined as follows: for any we have .
Let and be -sets. Recall that by -map from to we understand any function such that for any and we have .
It is easy to see that family of all -sets and -maps forms a category (with standard comoposition). We shall denote this category by . Moreover, by we shall denote full subcategory of whose objects are all finite -sets.
From -sets and one can construct another -set in two interesting (from our point of view) ways, i.e. by taking disjoint union with obvious group action and by taking product with group action as follows: Moreover it is clear that when and are finite, so are and .
Consider a finite -set . Then there exist a natural number , finite family of subgroups of and an isomorphism (in category)
Therefore (since is finite) family of isomorphism classes of forms a countable set.
Denote by the set of isomorphism classes of category . Then one can turn into a semiring as follows: for any finite -sets and define
Note that here we treat the empty set as a -set (with one and unique group action), therefore has zero element (the other way is to formally add the zero to - this is just technical thing).
Define by the Grothendieck group of . If is an abelian semigroup and is a bilinear map, then it can be uniquely extended to a bilinear map , therefore can be uniquely turned into a ring from . This ring is called the Burnside ring of .
each element of can be expressed as a formal diffrence ;
is a commutative, unital ring, where is the unity of ;
can be turned into a contravariant functor from the category of finite groups to the category of commutative, unital rings;
is a cancellative semigroup, therefore it embedds into ;
for the trivial group there is a ring isomorphism ;
for any group there is a ring monomorphism for some natural number ; this is called the characteristic embedding;
for any two groups we have: if and are isomorphic (as a rings), then ; generally need not be isomorphic to .
|Date of creation||2013-03-22 18:08:02|
|Last modified on||2013-03-22 18:08:02|
|Last modified by||joking (16130)|