Seminar of Graph Theory - Martin Vodička (12.4.2018)
Thursday 12.4.2018 at 9:50, Lecture room M/213
By: Martin Škoviera
Local embeddability of groups into finite loops
The concept of a group locally embeddable into finite groups (LEF-group) was introduced by Vershik and Gordon in 1998 to be a group where every finite square cut out of its multiplication table can be extended to a multiplication table of a finite group.
Glebsky and Gordon showed (2005) that a group is locally embeddable into finite semigroups if and only if it is an LEF-group, and that every group is locally embeddable into finite quasigroups, and even into finite loops. Improving on their result Ziman in 2005 proved that every group, as well as every loop having antiautomorphic two-sided inverses is locally embeddable into finite AAI~loops. However, the AAI loops are still ``rather far away from groups'' in general. A much closer class to groups is formed by the loops satisfying the so called inverse property. We show that every IP-loop, henceforth every group, is locally embeddable into finite IP-loops. The proof makes use of Steiner triple systems and Dirac's theorem for graphs containing a Hamilton cycle.