Abelian Extensions and Solvable Loops

作者：Stanovsky David^*; Vojtechovsky Petr

来源：Results in Mathematics, 2014, 66(3-4): 367-384.

DOI：10.1007/s00025-014-0382-6

• Commutator theory for loops
• abelian normal subloop
• solvable loop
• nilpotent loop
• abelian extension
• central extension

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.

• 出版日期2014-11