On The Structure Of The Augmentation Ideals And Quotient Groups For Integral Group Ring

Let G be a group, ZG its intergral group ring andΔ(G) the augmentationideal of ZG, denoteΔⁿ(G) the n'th power ofΔ(G) which is generated as anabelian group by all products (g₁-1)…(g_n-1), where g₁,…,g_n∈G\{1}. Theproblem of determining the structure ofΔⁿ(G) and quotient groupsQ_n(G)=Δⁿ(G)/Δⁿ⁺¹(G) is an interesting topic in group ring theory. For abeliangroups many works have been done, as for nonabelian groups much less is known sofar, in this paper we mainly consider nonabelian group case.In Chapter 1, we review briefly the developments of group ring theory, andintroduce the contents of this dissertation together with some elementary knowledgewhich are necessary in this paper.Dihedral groups are one kind of important class in nonabelian groups, inChapter 2 we classified them into D_2^tk(t≥0, k odd) by theirs orders divided by thelargest power of 2. In the following aset of Z-basis forΔⁿ(D_2^tk) is given fort=0 or t=1 respectively, so the structure of Q_n(D_2^tk) canbe determined easily.As for t≥1 we find out a relation betweenΔⁿ(D_2^tk) andΔⁿ⁺¹(D_2^tk), using thisrelation we give an upper bound 2t+1 for the rank of Q_n(D_2^tk) as an elementary2-group.For an elementary p-group G, a set of Z-basis forΔⁿ(G) is giveninductively by Parmenter in[46]. We generalize Parmenter's result in Chaper 3 to afinte direct sum of elementary p_i-group (P_i distinct). We also determine thestructure ofΔⁿ(G) and Q_n (G) for any group (abelian or not) G with order pq (p,q prime).We investigate the structure ofΔⁿ(G) and Q_n(G) for nonabelian group Gwith a perfect normal subgroup H in§4.2, and give a set of Z-basis forΔⁿ(G)when G/H is cyclic group or elementary p-group, this enables us to determinethe structure ofΔⁿ(S_m) and Q_n(S_m), where S_m is the m'th symtric group. Evenif ignoring the structure of G/H we still have Q_n(G)(?)Q_n(G/H), from this resultwe can convert the problem of determining the structure of quotient groups for anygroups to those solvable groups. In§4.3, for any nonabelian group G, we find a setof generators for Q_n(G) related to G's Sylow p_i-subgroups S_pi, based on thisgenerating set Q_n(G) can be decomposed into a direct sum as Q_n(G)=(?)Q'_n(S_pi),where Q'_n(S_pi)=Δⁿ(S_pi)/(Δⁿ(S_pi)∩Δⁿ⁺¹(G)) and s=|{p_i|p_i||G|}|. We also haveQ'_n(S_pi)=0 if and only if S_pi(?)[G, G], by using this and those previous resultsweshow Q_n(D_2^tk)(?)Q_n(D_2^t),so the structure of Q_n(D_2^tk) is closely retated to t.For a kind of p-group with N_p-series, a method for computing the rank ofQ_n(G) as elementary p-group is provided by G.Losey and N.Losey in[65]. Byusing this method in Chapter 5 we have Q_n(D₄)(?)Z₂⊕Z₂⊕Z₂⊕Z₂⊕Z₂ andalso prove a well-known result on the structure of Q_n(G) for elementaryp-group G which was obtained by Passi in[67]. At last we compute the rank ofQ_n(G) for some nonabelian p-groups with N_p-series.