Loops as invariant sections in groups, and their geometry. (English) Zbl 0814.20055

Nagy, Péter T.; Strambach, Karl

Can. J. Math. 46, No. 5, 1027-1056 (1994).

A deep and detailed investigation of the so-called left conjugacy closed loops (l.c.c. loops) is given. In such a loop the set of left transformations \(x \mapsto gx\) is invariant with respect to the inner automorphisms of the group \(G\) generated by the left transformations. Some examples of l.c.c. loops are constructed. The intersections of the class of such loops with some well-known loop classes are found. Also, properties of universal l.c.c. loops are described. It is proved, in particular, that there exist l.c.c. loops which are \(G\)-loops but not conjugacy closed. In detail, the Burn loops (i.e., l.c.c. loops with the left Bol property) are investigated. Some general theorems about differentiable l.c.c. loops are proved. In conclusion, the geometric properties of l.c.c. loops, that is the configurations on the corresponding 3-net associated with the loops, are considered.

Reviewer: A.M.Shelekhov (Tver’)

MSC:

20N05	Loops, quasigroups
22A30	Other topological algebraic systems and their representations
05B30	Other designs, configurations
51A99	Linear incidence geometry
53C30	Differential geometry of homogeneous manifolds

Keywords:

left conjugacy closed loops; left transformations; inner automorphisms; l.c.c. loops; \(G\)-loops; Burn loops; left Bol property; differentiable l.c.c. loops; 3-net