On the radical idealizer chain of symmetric orders

Abstract

If $Λ$ is an indecomposable, non maximal, symmetric order, then the idealizer of the radical $Γ:= \Id(J(Λ)) = J(Λ)^{#} $ is the dual of the radical. If $Γ$ is hereditary then $Λ$ has a Brauer tree (under modest additional assumptions). Otherwise $Δ:= \Id(J(Γ)) = (J(Γ)^2)^{#} $. If $Λ= \Z_p G$ for a $p$-group $G\neq 1$, then $Γ$ is hereditary iff $G\cong C_p$ and otherwise $[Δ: Λ] = 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 $G$.