On extensions of Moufang loops by a cyclic factor that is coprime to three

ABSTRACT

Gagola’s construction of cyclic extensions of Moufang loops for orders coprime to three is investigated with the goal of describing necessary and sufficient conditions when the construction really gives a Moufang loop. This is achieved both in terms of equations and loop autotopisms. The obtained description is then specialized to Moufang loops that cyclically extend a normal subgroup G, showing the connection to Chein’s construction in case when the semiautomorphism characterizing the extension is an antiautomorphism of G. These results are then illustrated upon the case when G is a noncommutative group that possesses a cyclic subgroup of index two and has a small center (i.e.|Z(G)|∈{1,2}).

KEYWORDS:

• Cyclic extension
• Moufang loop
• semiautomorphism
• semidirect product

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20N05