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\(p^ *\)-theory and modular representation theory. (English) Zbl 0618.20007
Let F denote a field of characteristic \(p>0\) and let G be a finite group. Let \(f_ 0(G)=\sum_{g\in G}a_ gg\) denote the primitive central idempotent of the principal b-block, \(B_ 0(G)\), of the group algebra FG. Set \(Supp(f_ 0(G))=\{g\in G|\) \(a_ g\neq 0\}\) and \(O_{f_ 0}(G)=<g|\) \(g\in Supp(f_ 0(G))>\), so that \(Supp(f_ 0(G))\) is a characteristic subset of G and \(O_{f_ 0}(G)\) is a characteristic subgroup of G. It is known that \(O_{f_ 0}(G)\) depends only on G and the characteristic p of F. This interesting paper demonstrates connections between \(O_{f_ 0}(F)\) and the generalized p’-core of G, \(O_ p*(G)\) [cf. H. Bender, Hokkaido Math. J. 7, 271-288 (1978; Zbl 0405.20015)]. The main results of the paper are:
Theorem 2.1: \(O_{f_ 0}(G)\) is a \(p^*\)-group. In particular, \(O_{f_ 0}(G)\leq O_ p*(G).\)
Theorem 2.2: \(O_{f_ 0}(N_ G(P))=O_{f_ 0}(C_ G(P))\leq O_{f_ 0}(G)\) for all p-subgroups P of G.
Theorem 2.6: If \(p\neq 2\), then \(O_{f_ 0}(G)=O_ p*(G).\)
The paper concludes by describing the necessary alterations for the \(p=2\) case in Remark 2.8.
Reviewer: M.E.Harris

MSC:
20C20 Modular representations and characters
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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