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In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in , introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory.
Let k be a field of characteristic p, where p is a prime number, let pp_k(G) be the Grothendieck group of p-permutation kG-modules, where G is a finite group, and let Cpp_k(G) be pp_k(G) tensored with
Let k be an algebraically closed field of characteristic p, where p is a prime number or 0. Let G be a finite group and ppk(G) be the Grothendieck group of p-permutation kG-modules. If we tensor it wi
The Grothendieck group of permutation modules for a finite group G becomes a biset functor when G is allowed to vary. All the composition factors of this biset functor are determined.