ar X iv : 0 90 1 . 30 39 v 1 [ m at h . G R ] 2 0 Ja n 20 09 SUBGROUPS OF DEPTH THREE AND MORE

A subalgebra pair of semisimple complex algebras B ⊆ A with inclusion matrix M is depth two if MM t M ≤ nM for some positive integer n and all corresponding entries. If A and B are the group algebras of finite group-subgroup pair H < G, the induction-restriction table equals M and S = MM t satisfies S 2 ≤ nS iff the subgroup H is depth three in G; similarly depth n > 3 by successive right multiplications of this inequality with alternately M and M t. We show that a Frobenius complement in a Frobenius group is a nontrivial class of examples of depth three subgroups. Depth-3 towers of Hopf algebras are also considered: a tower of Hopf algebras A ⊇ B ⊇ C is shown to be depth-3 if C ⊂ core(B).