Abelian extensions and solvable loops

TL;DR

This paper explores the structure of abelian extensions in loops, characterizing abelian subloops and their relation to solvability, and connects these algebraic properties to computational complexity.

Contribution

It provides a syntactic and semantic characterization of abelian normal subloops and links abelian extensions to congruence solvability in loops.

Findings

A loop is congruence solvable iff it is not Boolean complete.

Connections established between nilpotence and solvability in loops.

Highlights the analogy between central extensions and abelian extensions.

Abstract

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.