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On subgroup depth. (With an appendix by S. Danz and B. Külshammer). (English) Zbl 1266.20001

Summary: We define a notion of depth for an inclusion of complex semisimple algebras, based on a comparison of powers of the induction-restriction table (and its transpose matrix) and a previous notion of depth in an earlier paper of the second author. We prove that a depth two extension of complex semisimple algebras is normal in the sense of Rieffel, and conversely. Given an extension \(H\subseteq G\) of finite groups we prove that the depth of \(\mathbb CH\) in \(\mathbb CG\) is bounded by \(2n\) if the kernel of the permutation representation of \(G\) on cosets of \(H\) is the intersection of \(n\) conjugate subgroups of \(H\). We prove in several ways that the subgroup depth of symmetric groups \(S_n\subseteq S_{n+1}\) is \(2n-1\).
An appendix by S. Danz and B. Külshammer determines the subgroup depth of alternating groups \(A_n\subseteq A_{n+1}\) and dihedral group extensions.

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16D90 Module categories in associative algebras
16S20 Centralizing and normalizing extensions
16S34 Group rings
20B35 Subgroups of symmetric groups
20C15 Ordinary representations and characters
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