of H has order 2n, while the group N/H has order 2n–1. Examples In the proof of the Frobenius theory about the existence of a kernel of a Frobenius group G where the subgroup H is the subgroup fixing a point and S is the set of all irreducible characters of H, the isometry τ on I0(S) is just induction, although its extension to I(S) is not induction. Similarly in the theory of exceptional characters the isometry τ is again induction. In more complicated cases the isometry τ is no longer induction. For example, in the Feit–Thompson theorem the isometry τ is the Dade isometry. References Finite groups Representation theory