Limited Group Of A Given Defect Group P-block Of Existence

In what conditions a p-subgroup D of a finite group G can be a defect group for some p-block of G, the problem is important in the modular representation theory of finite groups. The present dissertation studies on the above problem, especially in the case D =< 1 >, i.e. on the existence of p-blocks with defect 0, we obtain some group theoretic conditions for the existence of blocks with defect 0 in a finite group.Applying a (g, D)-pair defined by Shengming Shi, firstly, by establishing the relationships between |Φ(g)| and the numbers of (g, D)-pairs, we give a sufficient and necessary condition for a finite group G having a p-block with defect group D. Secondly we establish the relationship between the sum of p-block idempotents of G with defect group D and the sum of the p-block idempotents of defect zero of G/D.Next we obtain some group theoretic conditions for the existence of p-blocks with defect 0 in some special groups, for instance, in an extension group of a nilpotent group by a abelian group, in a p-nilpotent group, in a group with all subgroups of order p are conjugate, and so on. Also under some special conditions, we investigate the existence of p-blocks with defect 0 of even order groups, and consequently we generalize the part results of Ito and Jiping Zhang.Finally, we establish the relationship between characters of a finite group G and that of a normal subgroup of G, especially on characters with p-defect 0.