Seminár z algebratickej teórie grafov - Martin Vodička (18.5.2018)

v piatok 18.5.2018 o 13:30 hod. v miestnosti M 213

16. 05. 2018 09.41 hod.
Od: Martin Mačaj

Prednášajúci: Martin Vodička

Názov: Local embeddability of groups into finite IP loops

Termín: 18.5.2018, 13:30 hod., M/213

In 1998 A.M.Veršik and E.I. Gordon introduced in their article notion of group locally embeddable into finite groups (LEF). Group is LEF if every finite square subtable of its multiplicative table can be extended into multiplicative table of some finite group.
It has been proven that not every group has this property, so natural step was to weaken structure which we want group embed into. It has been proven that every group is locally embeddable into finite quasigroups and finite loops. This result was improved by M. Ziman in 2005 when he proved that every group is locally embeddable into finite so called IAA loops. However, IAA loops are still quite "distant" from groups. Much closer to groups are IP loops which have the inverse property. We will prove that every IP loop, and thus also every group, is locally embeddable into finite IP loops.
Proof in constructive, it uses Steiner triple system and in the end we transform this problem on problem from graph theory whree we use also Dirac characterization of hamiltonian graphs. We will look more closely at the last (and probably most important) part of the proof.