Calculation of a K2 group of an F2 coefficients noncommutative group algebra
Abstract
In this paper, the K2 group of F2 coefficients group algebra of a noncommutative group with 8 elements(dihedral group D4 ) is calculated,which is divided into three parts:The first part is the introduction of basic knowledge related to algebra K-theory, and a method of Magurn to calculate finite field coefficients noncommutative finite group algebra in reference [2]. In the second part, operation laws of Dennis-Stein symbols is introduced, and we combined it with the fact that F2[D4] is a local ring to determind the direct sum term of K2(F2[D4]) can only be Z2 or Z4. In the third part, we continue to make use of the fact that F2[D4] is a local ring, and proved that the group D1(F2[D4]) is an abelian group closely related to the group K2(F2[D4]) through operating Dennis-Stein symbols. Then, we used group homology and the Kunneth formula of the finite abelian group version to calculate all cases of H2(D1(F2[D4]),Z) , and substituted the obtained results into the long exact sequence derived from the Hochschild-Serre spectral sequence for testing, and finally constructed the result: K2(F2[D4])=Z2..