Defect groups play a key role in the block theory of modular representations of finite groups and are the most important object which connects group theoretical properties and representation theoretical properties. It is well known that the problem-to decide when a given p-subgroup D is a defect group of some p-block of a finite group G-is of great importance in the modular representation theory of finite groups. This problem was posed as Problem 19 in [1] by Brauer and as Problem 5 in [3] by Feit. In 1983 Robinson [9] obtained a precise formula for the number of p-blocks of G with a given defect group D However, it is by no means easy to calculate the number according to this formula. So for a concrete p-subgroup D enumerating p-blocks of G with D as a defect group is still an important task. When D is a Sylow p-subgroup or a maximal subgroup of some Sylow p-subgroup, the condition for the existence of p-blocks of G with D as a defect group has been obtained (see [3, IV 4.17] and [11, Theorems 3 and A]).
In the present paper we still study the problem, and we consider the case of D being a submaximal subgroup of a Sylow p-subgroup (i.e., the case of * Research