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 ≠ 1 , then Γ is hereditary iff G ≅ 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 p G 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.