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Morita Equivalent Blocks in Non‐Normal Subgroups and p ‐Radical Blocks in Finite Groups
Author(s) -
Hida Akihiko,
Koshitani Shigeo
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799007139
Subject(s) - mathematics , quotient , normal subgroup , residue field , discrete valuation ring , combinatorics , discrete valuation , block (permutation group theory) , ideal (ethics) , finite group , order (exchange) , group (periodic table) , maximal ideal , field (mathematics) , finite field , group algebra , discrete mathematics , pure mathematics , algebra over a field , physics , quantum mechanics , philosophy , epistemology , finance , economics
Let O be a complete discrete valuation ring with unique maximal ideal J (O), let K be its quotient field of characteristic 0, and let k be its residue field O/ J (O) of prime characteristic p . We fix a finite group G , and we assume that K is big enough for G , that is, K contains all the ∣ G ∣‐th roots of unity, where ∣ G ∣ is the order of G . In particular, K and k are both splitting fields for all subgroups of G . Suppose that H is an arbitrary subgroup of G . Consider blocks (block ideals) A and B of the group algebras RG and RH , respectively, where R ∈{O, k }. We consider the following question: when are A and B Morita equivalent? Actually, we deal with ‘naturally Morita equivalent blocks A and B ’, which means that A is isomorphic to a full matrix algebra of B , as studied by B . Külshammer. However, Külshammer assumes that H is normal in G , and we do not make this assumption, so we get generalisations of the results of Külshammer. Moreover, in the case H is normal in G , we get the same results as Külshammer; however, he uses the results of E. C. Dade, and we do not.