These notes accompany a series of three lectures on automorphic loops to be delivered by the author at Workshops Loops ’15 (Ohrid, Macedonia, 2015). Automorphic loops are loops in which all inner mappings are automorphisms. The first paper on automorphic loops appeared in 1956 and there has been a surge of interest in the topic since 2010. The purpose of these notes is to introduce the methods ...

X iv :0 80 3. 04 72 v1 [ m at h. R A ] 4 M ar 2 00 8 On decomposition of commutative Moufang groupoids B.V.Novikov It is well-known that the multiplicative groupoid of an alternative/Jordan algebra satisfies Moufang identities [1, 4]. Therefore it seems interesting to study the structure of such groupoids. In this note we apply to Moufang groupoids an approach which is widespread in Semigroup T...

Introduction. If £ is a group with a normal subgroup K one may form the quotient group E/K^M. Conversely, for preassigned groups K, M, there is the extension problem: to determine (in some sense) all groups E with K as normal subgroup such that E/K^M. Much progress has been made on this problem, particularly through the work of Baer [l, 2, 3] and the cohomology theory of Eilenberg and MacLane [...

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Mouf...

Given a uniquely 2-divisible group $G$, we study commutative loop $(G,\circ)$ which arises as result of construction in ``Engelsche elemente noetherscher gruppen'' (1957) by R. Baer. We investigate some general properties and applications ``$\circ$'' determine necessary sufficient condition on $G$ order for $(G, \circ)$ to be Moufang. In ``A class loops categorically isomorphic Bruck odd order'...

It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM-and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple F-qua...

We coordinatize the Moufang generalized octagons arising from the Ree groups of type 2 F4. In this way, we obtain a very concrete and explicit description of these octagons. We use this to prove some results on suboctagons, generalized homologies, Suzuki-Tits ovoids and groups of projectivities of the Ree octagons. All our results hold for arbitrary Ree octagons, finite or not.