Means via groups and some properties of autodistributive Steiner triple systems

Abstract

The classical definition of arithmetical mean can be transferred to a group (G, +) with the property that for any y∈G there is a unique x∈G such that x + x = y (uni-2-divisible group). Indeed we define in G a commutative and idempotent operation ∇ that recalls the classical means, even if (G,+) is not commutative. Afterwards in section 3 we show that, by means of the commutative and idempotent operation ∇ usually associated with an autodistributive Steiner triple system (G,L), we can endow any plane of (G,L) of a structure of affine desarguesian (Galois) plane.