Relative \(\pi\)-blocks of \(\pi\)-separable groups (Q1818861)

It was shown by M. Slattery and others that there exists a reasonable theory of \(\pi\)-blocks of \(\pi\)-separable groups, for \(\pi\) a set of primes. Some parts of this theory are generalized in the paper under review. Given a \(\pi'\)-special character \(\mu\) of a normal subgroup \(N\) of a finite \(\pi\)-separable group \(G\), the irreducible characters of \(G\) lying over \(\mu\) are distributed into subsets called relative \(\pi\)-blocks with respect to \(\mu\). (In case \(N=1\) these coincide with the classical \(\pi\)-blocks.) The main result of the paper is a version of the Fong reduction for relative \(\pi\)-blocks. Defect groups of relative \(\pi\)-blocks are defined, and some applications to relative character heights are given. Reviewer's remark: In the examples at the end of the paper the relative \(\pi\)-blocks seem to coincide with the classical \(\pi\)-blocks of the factor group; so they do not illustrate the new theory too well.