Authors: 

M.~Hertweck and W.~Kimmerle

Universit\"at Stuttgart, Pfaffenwaldring 57, 
D-70550 Stuttgart, Germany 

Fax: (49 711)685-5304

Title:

On principal blocks of $p$-constrained groups

Abstract:

A theorem of K.~W.~Roggenkamp and L.~L.~Scott \cite{RoSc:87b} 
shows that for a finite group $G$ with a normal $p$-subgroup containing 
its own centralizer, each two group bases of the integral group ring $\ZZ G$ 
are conjugate in the units of $\ZZ_{p}G$.
Though the theorem presents itself in the work of others and appears
to be needed (cf.~\cite{kim:92c,KiRo:93a,KiRo:93b,sco:87,sco:90}),
there is no published account. 
There seems to be a flaw in the proof, because 
a `theorem' appearing in the surveys \cite{rogg:91b,rogg:91c},
where the main ingredients to a proof are given, is false. 
In this paper, it is shown how to close this gap, at least if
one is only interested in the conclusion mentioned above.
Therefore, some consequences of the results of A.~Weiss on 
permutation modules \cite{wei:87,wei:93} are stated.
The basic steps of which any proof should consist of are discussed in some
detail. In doing so, a complete, yet short proof of the theorem is given in 
the case that $G$ has a normal Sylow $p$-subgroup. 

Current status: submitted for publication