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Nilpotent decomposition in integral group rings
Tijdschriftbijdrage - Tijdschriftartikel
A finite group G is said to have the nilpotent decomposition property (ND) if for every nilpotent element α of the integral group ring Z[G] one has that αe also belong to Z[G], for every primitive central idempotent e of the rational group algebra Q[G]. Results of Hales, Passi and Wilson, Liu and Passman show that this property is fundamental in the investigations of the multiplicative Jordan decomposition of integral group rings. If G and all its subgroups have ND then Liu and Passman showed that G has property SSN, that is, for subgroups H, Y and N of G, if N◁H and Y⊆H then N⊆Y or YN is normal in H; and such groups have been described. In this article, we study the nilpotent decomposition property in integral group rings and we classify finite SSN groups G such that the rational group algebra Q[G] has only one Wedderburn component which is not a division ring.