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 induction-restriction table M (and its transpose matrix). This notion of depth coincides with the depth from [Kadison, 2008]. In particular depth 2 extensions coincides with normal extensions as introduced by Rieffel in 1979. 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 S_n < S_{n+1} is 2n-1. An appendix by S. Danz and B. Kuelshammer determines the subgroup depth of alternating groups A_n < A_{n+1} as well as dihedral groups.
Keywords
Cite
@article{arxiv.0906.0440,
title = {On subgroup depth},
author = {Sebastian Burciu and Lars Kadison and Burkhard Kuelshammer},
journal= {arXiv preprint arXiv:0906.0440},
year = {2010}
}
Comments
33 pp, new appendix by S. Danz and B. Kuelshammer, where the depth of the inclusion of alternating groups A_n < A_{n+1} is determined