On the radical idealizer chain of symmetric orders
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=ZpG for a p-group G≠1, then Γ is hereditary iff G≅Cp and otherwise [Δ:Λ]=p2|G/(G′Gp)|. For Abelian groups G, the length of the radical idealizer chain of ZpG is (n−a)(pa−pa−1)+pa−1, where pn is the order and pa the exponent of the Sylow p-subgroup of G.