Titel: On the radical idealizer chain of symmetric orders.

Author: Gabriele Nebe, Abteilung Reine Mathematik, Universit\"at Ulm, 89069 Ulm, Germany

Status: Preprint.

Abstract:
If $\Lambda $ is an indecomposable, non maximal,
 symmetric order, then the idealizer of the
radical $\Gamma := \Id(J(\Lambda )) = J(\Lambda )^{\#} $ is the dual of the
radical. If $\Gamma $ is  hereditary 
then $\Lambda $ has a Brauer tree
(under modest additional assumptions).
Otherwise $\Delta := \Id(J(\Gamma )) = (J(\Gamma )^2)^{\#} $.
If $\Lambda = \Z_p G$ for a $p$-group $G\neq 1$, then $\Gamma $ is hereditary 
iff $G\cong C_p$ and otherwise  $[\Delta : \Lambda ] = p^2 | G/(G'G^p)| $.
For Abelian groups $G$, the length of the radical idealizer chain 
of $\Z_pG$ is $(n-a)(p^{a} - p^{a-1})+p^{a-1}$, where $p^n$ is the order and
$p^a$ the exponent of the Sylow $p$-subgroup of $G$.