Calculation of a K_2 group of an F_2 coefficients noncommutative group algebra

TL;DR

This paper calculates the K_2 group of the group algebra of the dihedral group D_4 over the field F_2, using algebraic K-theory, Dennis-Stein symbols, and spectral sequences, revealing it is isomorphic to Z_2.

Contribution

It provides a detailed computation of K_2 for a specific noncommutative group algebra over F_2, applying advanced algebraic methods and spectral sequences.

Findings

K_2(F_2[D_4]) is isomorphic to Z_2

The direct sum term of K_2 can only be Z_2 or Z_4

D_1(F_2[D_4]) is an abelian group related to K_2(F_2[D_4])

Abstract

In this paper, the K_2 group of F_2 coefficients group algebra of a noncommutative group with 8 elements(dihedral group D_4 ) 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 F_2[D_4] is a local ring to determind the direct sum term of K_2(F_2[D_4]) can only be Z_2 or Z_4. In the third part, we continue to make use of the fact that F_2[D_4] is a local ring, and proved that the group D_1(F_2[D_4]) is an abelian group closely related to the group K_2(F_2[D_4]) 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 H_2(D_1(F_2[D_4]),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: K_2(F_2[D_4])=Z_2..