A Free Resolution On ZQ_4n And 3-Cohomology Theory

The subject of group cohomology began in the 1920s and has made a relatively comprehen-sive development in the late 1940s.It is still an active research field today.Group cohomology is one of the most important mathematical tools in the study of groups and is an important part of group theory,and computing cohomology is an important part of cohomology theory.According to homology theory,the cohomology group of a group is determined by the cocycle and coboundary of the Bar-resolution,but it is difficult to compute the cocycle and coboundary of the Bar-resolution of a general group,which makes the calculation of the cohomology group very difficult.Let G be a group.By a linear Gr-category over G we mean a tensor category VecG? consists offinite-dimensional vector spaces graded by G with the usual tensor product and with associativity constraint given by a 3-cocycle ? on G.A student Hoang Xuan Sinh of Grothendieck in her thesis has given the monoidal structures of VecG(the category of G-graded spaces)were first related to the third cohomology group of G.Gr-categories are a typical class of fusion categories,in particular,any pointed fusion category has the form VecG? and so does the full subcategory of semi-simple objects of any finite pointed tensor category.See[9],[10].The structure of tensor category VecG? under the condition of category equivalence is mainly determined by the 3-cohomology group of a group.The characterization of braided tensor category needs to be realized by the 3-cocycle of the group.At present,the relevant results are as follows:in the reference[1]and[2],the structure of 3-cohomology group on a cyclic group is given;in the reference[3],Bulacu,D.,Caenepeel,S.,and Torrecillas,B.and others studied cohomology of the noncyclic group:Klein group for the first time.The method is to calculate cohomology of Klein group by calculating very complex happy 3-cocycle,however,this method does not seem to be suitable for more general groups;in the reference[4],the 3-cocycle and 3-cohomology group of direct product of two arbitrary finite cyclic groups is given by HUANG Hualin,LIU Gongxiang and YE Yu.Their idea is to use free resolution on a group(minimal resolution).Firstly,free resolution(minimal resolution)on groups is found,and the 3-cocycle,3-coboundary and 3-cohomology of free resolution is calculated.Then,the chain map from Bar-resolution to free resolution is given.Lastly,The structures of 3-cocycle and 3-cohomology group on this group are given by means of the homology structure on the free resolution.In this paper,we generalize the structures of 3-cocycle and 3-cohomology on abelian groups to nonabelian groups.This paper aims to study the free resolution on dicyclic group(denoted as Q4n)and its homology structure:3-cocycle,3-coboundary and 3-cohomology group.There are three parts:firstly,a free resolution on Q4n is constructed,and the 3-cocycle,3-coboundary and 3-cohomology were given.Secondly,the chain map from Bar-resolution to free resolution of the Q4n group is established,and the commutative graph is proved,then it is obtained that the 3-cohomology group H3(Q4n,k*)of Q4n is a cyclic group of order 4n;Finally,we find a chain map from the free resolution of a group to the Bar-resolution,and give the corresponding relations between the 3-cocycle of the dihedral group D2n and Q4n.