Seminar 11/12/15: Finite k-nets in projective planes

Finite k-nets in projective planes
Nicola Pace
Dublin Institute of Technology
Friday 11 December 2015
1pm, Blue Room, DIT Kevin Street


This talk deals with k-nets embedded in the projective plane PG(2,K) defined over a field K. They are line configurations in PG(2,K) consisting of k≥3 pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component.

The case k=3 is particularly interesting because, in some instances, it is possible to realise finite groups. A 3-net is said to realise a group (G,·) when the following condition holds. If A,B,C are the components, then there exists a triple of bijective maps from G to (A,B,C), say α∶ G→A,   β: G →B,   γ: G → C, such that a·b =c if and only if α(a),β(b),γ(c) are three collinear points, for any a,b,c∈G. If K has zero characteristic, 3-nets realising a finite group are classified. If the characteristic of the field is p>|G|, then the same classification holds true apart from three possible exceptions: A4,S4 and A5.

Key ideas and results needed for the classification are presented.