Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

A subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

Primitive characters of maximal subgroups of M-groups

Let [Formula: see text] be a maximal subgroup of an [Formula: see text]-group [Formula: see text] with odd index and let [Formula: see text] be primitive. Lewis proved in this situation that [Formula: see text] divides [Formula: see text], and Isaacs and Wilde further refined this result by showing that either [Formula: see text] or [Formula: see text]. In this paper, we present an independent and simpler proof for these remarkable results and thereby obtain more detailed information regarding the structure of the group [Formula: see text] and the primitive character [Formula: see text]. In particular, [Formula: see text] is strongly irreducible in the sense of Brauer.

Spectral Flow Argument Localizing an Odd Index Pairing

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.