The Loewy layers of principal indecomposable modules
Let \(G\) be a \(p\)-solvable group, and let \(k\) be the field of \(p\) elements. The author obtains results about the Loewy layers of the principal indecomposable module \(P_ 1\) corresponding to the trivial module \(k\). For \(G\) a \(p\)-group the structure of the Loewy layers of \(P_ 1 = kG\) is completely described by the Jennings polynomial. Here the author gives a generalization of this polynomial to \(p\)-solvable groups where the situation is much more complicated: For the construction of the polynomial a lot of information about the group and its module theory is needed, also the polynomial does not yield the structure of the Loewy layers in a straightforward way. The specific applications given in this paper include a new proof of the celebrated theorem of Gaschütz about \(P_ 1J/P_ 2J\), a new proof of the result of Green-Hill on the occurrence of the chief factors of \(G\) as composition factors of \(P_ 1\), a complete description of the Loewy layers in case \(G\) is of \(p\)- length 1, and a description of the layer \(P_ 2J/P_ 3J\) in case \(G\) is of \(p\)-length 2.
scientific article; zbMATH DE number 130535