Über Einheitengruppen modularer Gruppenalgebren

Wirsing, Sven

We analyse unit groups E(KG) of group algebras KG for non-abelian p-groups G and fields K of characteristic p.By calculating the core and the normaliser in 1+rad(KG) for every subgroup U of G, we generalise results of Pearson and Coleman.Our concept of so-called " end-commutable ordering " leads to a new method of studying the center of 1+rad(KG). We proof that a finite group G is nilpotent if and only if every conjugacy class possesses an end-commutable ordering. As a simple consequence we get a result of Bovdi and Patay, which shows how the exponent of Z(1+rad(KG)) may be determined by calculations purely within the group G. We describe the groups for which this exponent is extremal. We calculate the exponent for wreath products and for central products. Another application of our concept of end-commutable ordering is a description of the invariants of Z(1+rad(KG)) (for a finite field K). As a consequence of our results we prove that Z(1+rad(KG)), (1+rad(KG))' and (1+rad(KG))^p are not cyclic. Furthermore, we obtain some properties of unit groups of group algebras for extra-special 2-groups and fields of characteristic 2.

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Wirsing, Sven: Über Einheitengruppen modularer Gruppenalgebren. 2005.

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