Projective Indecomposable Characters And Their Modules

Projective indecomposable character is one of the most important conceptsin the modular representation theory of ?nite groups. In this thesis, some proper-ties of projective indecomposable characters of a ?nite group are given, includingsome relations about the degrees of projective indecomposable characters of a?nite group and its normal subgroups and the correspondence of the irreducibleconstituents of projective indecomposable characters of a ?nite group and thenormalizer of a Sylow p-subgroup. Furthermore, the higher Frobenius-Schur indi-cators of projective indecomposable characters of a ?nite group are investigated.As a special case of projective indecomposable characters, projective irre-ducible characters of a ?nite group play an extremely important role in represen-tation theory of ?nite groups. As one will see, the fact that most ?nite nonabeliansimple groups have projective irreducible characters leads to a characterizationof ?nite nonsolvable groups whose character graphs have no triangle and a clas-si?cation of ?nite groups having exactly two p-blocks under certain assumptions.Finally, this thesis explores an application of projective indecomposable char- acters inπ-theory forπ-separable groups and shows aπ-form of a conjecture ofProf. Willems.