On Block Induction

Block induction is a correspondence between certain p-blocks of subgroups and blocks of the whole group. It is a tool with the help of which one often can reduce the solution of problems in the whole group to smaller groups. The literature lists several ways to define induced blocks (in the sense of Brauer, Alperin and Burry, Green, and Wheeler). It is well-known that any two of these concepts of block induction coincide in their common domain of definition; for example, each of them is defined if the subgroup H of the group G satisfies DCG(D) ≤ H ≤ NG(D), for a p-subgroup D of G. We study some other features of blocks which are compatible with the above induction concepts in this sense, and formulate them as induction concepts so that one could deal with these and the above concepts in a unified way. We compare their fields of definition to the others in general and in the special cases of defect zero blocks and of p-groups. We give some sufficient conditions so that induction would not be defined in any sense. We show that, unlike in the case of Brauer sense induction, block induction in the sense of Alperin-Burry is not always defined from the normalizer of a chain of p-subgroups. Our example also shows that “admissible induction” and “being of multiplicity one” are not transitive. In the last part of the paper, we study the connection of the defect groups of the induced and the original block, in the case where the latter group is abelian.