Moufang loops of small order. I

Abstract

The main result of this paper is the determination of all nonassociative Moufang loops of orders <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to 31"> <mml:semantics> <mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mn>31</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq 31</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange’s theorem and Sylow’s main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders <italic>p</italic> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <italic>pq</italic> (for <italic>p</italic> and <italic>q</italic> prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2.