Exceptional smooth Bol loops.
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].
A loop is said to be left conjugacy closed if the set is closed under conjugation. Let be an LCC loop, let and be the left and right multiplication groups of respectively, and let be its inner mapping group, its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism determined by . In this short note we examine different possible extensions of this and the uniqueness of these extensions.
A set Q is a faithful homogeneous space over a commutative group iff there is a family S of mappings such that (Q,S) is a TST-space.
The study of paramedial groupoids (with emphasis on the structure of simple paramedial groupoids) was initiated in [1] and continued in [2], [3] and [5]. The aim of the present paper is to give a full description of finite simple zeropotent paramedial groupoids (i.e., of finite simple paramedial groupoids of type (II)—see [2]). A reader is referred to [1], [2], [3] and [7] for notation and various prerequisites.
In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, J. Algebra 317 (2007), 435–461, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the class of (pointed) F-quasigroups and the class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.
In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally,...