Research on defect groups of blocks of a group ring

Blocks of defect zero are easily understood objects in modular representation theory，since they are isomorphic to full matrix rings over the underlying field.Brauer‘s“Problem19”asks for group-theoretic criteria to determine when a group has such blocks.In connection with this problem，we proved the following result：let G be a solvable p-nilpotent group for some prime p，such that O y D2p i D2(G)=1.If certain groups is not involved in G，an element in O I D2p‘ii D2(G)with a centraliser of p’-order is found.Hence G has a p-block of defect zero.Next we proved the following result：Let G be a finite solvable group and p，a prime。Then G has a character of p-defect 0 if and only if G has a chain of subgroups G？？D21-D2？？G-D-22-D2？？G I D2n个D2，which satisfies the some conditions.By using the theorem，we can investigate without charactertable whether given groups have blocks of defect zero。

Blocks of defect zero are easily understood objects in modular representation theory, since they are isomorphic to full matrix rings over the underlying field. Brauer's "Problem 19" asks for group-theoretic criteria to determine when a group has such blocks. In connection with this problem, we proved the following result : let G be a solvable p-nilpotent group for some prime p, such that OィイD2pィエD2(G) = 1. If certain groups is not involved in G, an element in OィイD2p'ィエD2(G) with a centraliser of p'-order is found. Hence G has a p-block of defect zero.Next we proved the following result : Let G be a finite solvable group and p, a prime. Then G has a character of p-defect 0 if and only if G has a chain of subgroups G ⊇ GィイD21ィエD2 ⊇ GィイD22ィエD2 ⊇ … ⊇ GィイD2nィエD2, which satisfies the some conditions. By using the theorem, we can investigate without charactertable whether given groups have blocks of defect zero.