On block induction
by
Erzsébet Horváth
Dept. Algebra, Faculty of Sciences, Technical University of Budapest
Coauthors: Thomas Breuer

Block induction is a correspondence between certain blocks of subgroups and blocks of the whole group. It is a tool with the help of which one very often can reduce the solution of certain problems in the whole group for smaller groups. The literature lists several ways to define induced blocks (in the sense of Brauer, Alperin and Burry, Blau and Wheeler). It is well known that any two of the above concepts of block induction coincide in their common domain of definition. 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 them in a unified way. We compare their fields of definition to the others' in general and also in the special case 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, induction in the sense of Alperin-Burry is not always defined from normalizers of chains of p-subgroups. Our exapmle also shows that for blocks admissible induction and being of multiplicity one are not transitive. We also study connection of the defect group of the induced block and the original block, in the case when the first group is abelian.

Date received: June 27, 2000