3-Nets realizing a group in a projective plane

Korchmáros, G. and Nagy, Gábor Péter and Pace, N. (2014) 3-Nets realizing a group in a projective plane. JOURNAL OF ALGEBRAIC COMBINATORICS, 39 (4). pp. 939-966. ISSN 0925-9899

Preview

Text 1104.4439.pdf Available under License Creative Commons Attribution. Download (289kB) | Preview

Abstract

In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds true in characteristic p>0 apart from three possible exceptions Alt_4, Sym_4 and Alt_5.

Actions (login required)

Edit Item