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Chapter 2 Restriction to Normal Subgroups

DOI: 10.1016/s0304-0208(08)70228-7
  • Mathematics


Publisher Summary This chapter examines the process of restriction of irreducible and indecomposable modules to normal subgroups. After introducing some basic properties of induced and relatively projective modules, a classical theorem of Clifford is proved. The treatment is augmented by relating the process of restriction to that of the extension of ground fields. The chapter provides some important results pertaining to lifting of idempotents, including a theorem due to Thévenaz. As an application, the process of restriction of indecomposable modules to normal subgroups is examined. A similarity between seemingly unrelated procedures, namely between restriction of modules to normal subgroups and decomposition of modules under ground field extension is established. It turns out that the role of a conjugate of a module is played by a Galois conjugate, while the Schur index plays the role of the ramification index. The chapter considers some relations between VN and P (V) , where V is an irreducible FG-module, N a normal subgroup of G and P(V) is a projective cover of V.

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