Nearfields

New Features:

• This release includes a new package implementing finite nearfields and related finite projective (non-Desarguesian) projective planes.

  A nearfield satisfies all the axioms of a field except for the commutative law of multiplication and one of the distributive laws. In the Magma implementation the nearfields satisfy the right distributive law.

  • The finite sharply doubly transitive permutation groups are in one-to-one correspondence with the finite nearfields.
  • Nearfields coordinatise a class of translation planes.
  • Nearfields and they are the starting point for the construction of the Hughes planes.
• The function DicksonNearfield(q,v) creates a Dickson nearfield from the Dickson pair (q, v) , and the function ZassenhausNearfield(n) create one of the 7 exceptional nearfields.
• The group of units of a nearfield is metacyclic: it is available as a matrix group, a permutation group and as PC-group.
• The function AffineGroup(N) returns the semidirect product of the additive group of N by the group of units. All sharply doubly transitive groups occur in this form.
• Nearfields can be tested for isomorphism and the full automorphism group of a nearfield has been implemented.
• The function ProjectivePlane(N) returns the non-Desarguesian projective plane coordinatised by the nearfield N .
• The function HughesPlane(N) returns the Hughes plane obtained from N . Note that N can be either a Dickson nearfield or a Zassenhaus nearfield. (In the literature, the planes obtained from Zassenhaus nearfields are often called generalised Hughes planes.)