Unit Groups And Morphic Problem Of Finite Group Algebra FG

The subject matter of this paper lies at ring theory, group theory and elementary number theory.The theory of finite ring is a very active area which is not only of great theoretical interest in itself but also found important applications both within mathematics (e.g., in combinatorics) and within the Engineering Sciences (in particular, in Coding Theory). In recent 30 years, due to the development of computer technology and internet, Coding Theory develops rapidly. Hence, the study about finite rings is promoted.Group algebra are very interesting algebraic structures. The unit groups of the group algebra RG plays a very important role in studying the relation between the structure of G and its group ring RG. Many authors and scholars focus on the unit group of integral group ring, however there little research on the unit group of group ring over field. For the group algebra FG, the characteristic of the field F does divide the order of G, then the group ring FG is semisimple, and we can have the unit group of FG by find the decomposition of FG. The structure of u(FA4),u(FS3),u(FS4) and u(FD10) are determined by R. K. Sharma, J. B. Srivastava and M. Khan in 2007 and 2009. When the characteristic of F divide the order of G, J. Gildea and L. Greedon begin to use the isomorphism between FG and a certain subgroup of the n x n matrices over F and other techniques to establish the structure of u(FG) from 2008. They established the structure of u(F3k D6),u(F5k D20),u(F2k D8),u(F3k(C3×D3))and u(F2k G),where G are the Non-Abelian groups of order 16 and exponent 4.In Chapter 1 of this paper,we summarize the history of the unit group of group ring and the morphic group ring. Besides,we give the notations and basic results of ring theory and morphic group ring.In Chapter 2,we determine the structure of the unit groups of the group algebra FG,where G is the group with order 21,G1=C3×C7 is a Abelian group,and G2=is non-Abelian groups of order 21.In Chapter 3,we completely determine the structures of u(F3k D12),u(F3k Q12), u(F2k D12)and u(F2k A4)where D12=is the dihedral group of order 12,Q12=is the gen-eralized quaternion group of order 12 and A4=is the alternating group of order 12.We established the struc-ture of u(F3k D12)to be(C36k(?)C32k)(?)C3k-14.When k is even,then u(F3k Q12)≌(C36k(?)C32k)(?)C3k-14;If k is odd,then u(F3k Q12)≌(C36k(?)C32k)(?)(C9k-1×C3k-12). u(F2k D12)≌C26k(?)(C2k×C2k-1×GL(2,△)),where△={r(g+g-1)|r∈F2k,g∈D6}.When k is even,u(F2k A4)≌((C2k×C42k)(?)C42k)(?)C2k-13;when k is odd, u(F2k A4)≌((C2k×C42k)(?)C42k))(?)(C4k-1×C2k-1).In the last chapter,Chapter 4,we study the morphic problem of ZnQ8 and ZnD8,where Q8=is quaternion groupm, D8=is the dihedral group of order 8.When n is odd,then ZnD8 and ZnQ8 are all morphic group ring.