Vol. 38, No. 1, 1971

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Blocks and F-class algebras of finite groups

William Francis Reynolds

Vol. 38 (1971), No. 1, 193–205

For an arbitrary field F of characteristic p 0, the usual partitioning of the p-regular elements of a finite group G into F-classes (F-conjugacy classes) is extended to all of G in such a way that the F-classes form a basis of a subalgebra Y of the class algebra Z of G over F. The primitive idempotents of E FY , where E is an algebraic closure of F, are the same as those of Z. By means of this fact it is shown that if p > 0 the number of blocks of G over F with a given defect group D is not greater than the number of p-regular F-classes L of G with defect group D such that the F-class sum of L in Z is not nilpotent; equality holds if Op,p,p(G) = G or if D is Sylow in G. The results are generalized to arbitrary twisted group algebras of G over F.

Mathematical Subject Classification 2000
Primary: 20C05
Received: 10 October 1970
Published: 1 July 1971
William Francis Reynolds