Rigid automorphisms of linking systems

A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

Characteristic classes of involutions in nonsolvable groups

Let {G,D_{0},D_{1}} be finite groups such that {D_{0}\trianglelefteq D_{1}} are groups of automorphisms of G that contain the inner automorphisms of G. Assume that {D_{1}/D_{0}} has a normal 2-complement and that {D_{1}} acts fixed-point-freely on the set of {D_{0}} -conjugacy classes of involutions of G (i.e., {C_{D_{1}}(a)D_{0}<D_{1}} for every involution {a\in G} ). We prove that G is solvable. We also construct a nonsolvable finite group that possesses no characteristic conjugacy class of nontrivial cyclic subgroups. This shows that an assumption on the structure of {D_{1}/D_{0}} above must be made in order to guarantee the solvability of G and also yields a negative answer to Problem 3.51 in the Kourovka notebook, posed by A. I. Saksonov in 1969.

Normal subgroups of invertibles and of unitaries in a C*-algebra

We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a {\mathrm{C}^{*}}-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple \mathrm{C}^{*}-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital \mathrm{C}^{*}-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.

Quantum holonomies and the Heisenberg group

Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements of the Heisenberg group [Formula: see text]. Group composition in [Formula: see text] corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group [Formula: see text] of [Formula: see text], making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of [Formula: see text] adjust these signed areas, and the discrete symplectic transformations of [Formula: see text] generate the modular group of [Formula: see text].

On Asano’s theorem

Let [Formula: see text] be a division algebra over an infinite field [Formula: see text] such that every element of [Formula: see text] is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every noncentral subspace of [Formula: see text] invariant under all inner automorphisms induced by algebraic elements contains [Formula: see text], the additive subgroup of [Formula: see text] generated by all additive commutators of [Formula: see text]. From the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that [Formula: see text] is generated by multiplicative commutators as a vector space over the center of [Formula: see text].