On the radical idealizer chain of symmetric orders

Abstract

If $\Lambda$ 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 $\Gamma$ is hereditary then $\Lambda$ has a Brauer tree (under modest additional assumptions). Otherwise $Δ:= \Id(J(Γ)) = (J(Γ)^2)^{#} $. If $\Lambda ={\mathbb{Z}}_{p}G$ for a $p$-group $G\ne 1$, then $\Gamma$ is hereditary iff $G\cong C_p$ and otherwise $[\Delta :\Lambda ]=p^2|G/(G^{\prime }{G}^p)|$. For Abelian groups $G$, the length of the radical idealizer chain of ${\mathbb{Z}}_{p}G$ 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 $G$.

Cited by in corpus (1)

• On blocks with cyclic defect group and their head orders