On subgroup depth

Abstract

We define a notion of depth for an inclusion of multimatrix algebras B ⊆ A based on a comparison of powers of the inductionrestriction table M (and its transpose matrix). This notion of depth coincides with the depth from [16]. In particular depth 2 extensions coincide with normal extensions as introduced by Rieffel in [20]. For a group extension H ⊂ G a necessary depth n condition is given in terms of the core of H in G. We prove that the subgroup depth of symmetric groups Sn < Sn+1 is 2n − 1. An appendix by S. Danz and B. Külshammer determines the subgroup depth of alternating groups An < An+1 and of some other group extensions.Remark A.6. Note that the ring extensions CS 1 ⊂ CS 2 and CA 2 ⊂ CA 3 are clearly of depth 2, since S 1 ✂ S 2 and A 2 ✂ A 3 .