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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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