Endo-permutation modules in p-solvable groups (Q1825954)

Some answers are given to the following question. If \(\chi\) is an irreducible ordinary character of the finite p-solvable group G, does there always exist an RG-lattice affording \(\chi\) for which the sources are endo-permutation modules? In general, the answer is ``no''. The group \(GL_ 2(3)\) yields a counterexample for \(p=2\). On the other side, the main result of this article asserts: Suppose that G is p-solvable and that \(\chi\) is a character of G which is p-modularly irreducible. Then the sources of the unique (up to isomorphism) RG-lattice affording \(\chi\) are algebraic endo-permutation modules unless \(p=2\) and \(SL_ 2(3)\) or \(Z_ 5\cdot (Q_ 8\circ D_ 8)\) are involved in G. The rather technical proof uses a new result on the Clifford theory of tensor induction.