Hirosi Nagao

On multiply-transitive groups V

NOTATION. The stabilizer of points i, j ,..., k in G is denoted by G,j..., . On the other hand, Gtii...kl will denote the stabilizer in G of a set (i, j,..., k} of points; i.e., the subgroup of G consisting of all the elements fixing the set. If X is a subset of G fixing a subset d of 52, then X induces a set of permutations on A, which we denote by XA. For the subset X of G, I(X) will denote the set of all the fixed points of X. For a subgroup H of G we denote the normalizer of H in G by No(H), the centralizer of H in G by Co(H). The subgroup of G generated by the union of subsets S, T,... will be denoted by (S, T,...>.

Representations of Groups

This chapter presents the fundamental theory of representations of finite groups, including modular representations. It discusses ordinary representations. By an ordinary representation of G , it means a representation over a field K of characteristic zero, and the character defined by it is called an ordinary character. Such a representation is always completely reducible and determined uniquely up to equivalence by the character. All representations considered are taken over the complex field C . In particular, every irreducible representation is absolutely irreducible. The chapter also reviews modular representation theory ( p ). By a ( p -) modular representation, one means a representation of a group over a field of characteristic p > 0, and a character defined by such a representation is called a modular character. Some elementary facts on p -modular representations are established in the chapter for a fixed prime number p .

Rings and Modules

This chapter presents the fundamental results on rings and modules. It discusses a theorem that if A is a Noetherian (an Artinian) ring, then every finitely generated A -module is Noetherian (Artinian). In particular, if A is a finitely generated algebra over a commutative Noetherian (Artinian) ring R, then A is right and left Noetherian, (Artinian). The chapter provides an overview on idempotents and direct sum decompositions of ideals. It does not include a primitive idempotent decomposition of an idempotent, nor is it uniquely determined even if it did exist. It also reviews Krull–Schmidt–Azumaya theorem, which proves modules with composition series. The chapter explains that an integral domain R is said to be a Dedekind domain if it satisfies the three conditions: (1) R is a Noetherian ring, (2) R is integrally closed, and (3) any nonzero prime ideal of R is maximal.

Algebras and Their Representations

This chapter presents some fundamental facts about representations of algebras. It also presents theories of simple algebras, Schur indices, crossed products, and Frobenius algebras. An A -module that is finitely generated and free as an R -module is referred to as a representation module of A . Three conditions on the K -algebra A are equivalent. (1) A is separable, (2) A e is semisimple, and (3) A is A e -projective. The chapter also presents the studies of Schur index from the representation theoretic point of view.

4 – Indecomposable Modules

This chapter presents Greens theory on indecomposable modules of group rings over complete discrete valuation rings. Our first goal is the vertex theory of indecomposable modules in Section 3, which is fundamental throughout the theory. The Green correspondence in Section 4 is one of the main themes of this chapter. In this connection, we show a result due to Burry and Carlson (Theorem 4.6) and establish some functorial properties of it (Theorems 4.5 and 5.4). In Section 6 we study the endomorphism rings of induced modules and show a Clifford-type theorem on indecomposable modules (Corollary 6.8). The Green indecomposability theorem in Section 7 is one of the main results of this chapter, and in the proof of it, Theorem 13.27 of Chapter 1 concerning the extension of valuation will play an important role. In Section 8 we discuss Scott modules.

Theory of Blocks

This chapter presents Brauers first, second, and third main theorems on blocks, Fongs theory of block covering, and related results. For every principal indecomposable RH (respectively, FH )-module W belonging to b , there exist a principal indecomposable RG (respectively, FG )-module V such that W/V H . Principal indecomposable RH -modules are in one-to-one correspondence to principal indecomposable FH -modules via the reduction mod π. Thus, it suffices to prove the assertion in the case where W is a principal indecomposable FH -module.