Nilpotency in automorphic loops of prime power order

TL;DR

This paper proves that all commutative automorphic loops of odd prime power order are centrally nilpotent and constructs automorphic loops of order p^3 with trivial center using anisotropic planes.

Contribution

It establishes the nilpotency of commutative automorphic loops of odd prime power order and introduces a construction of loops with trivial center using geometric methods.

Findings

Commutative automorphic loops of odd prime power order are centrally nilpotent.

Constructed automorphic loops of order p^3 with trivial center.

Used anisotropic planes in matrix vector spaces for loop construction.

Abstract

A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with anisotropic planes in the vector space of $2\times 2$ matrices over the field of prime order $p$, we construct a family of automorphic loops of order $p^3$ with trivial center.