Moufang Loop - Examples

Examples

• Any group is an associative loop and therefore a Moufang loop.
• The nonzero octonions form a nonassociative Moufang loop under octonion multiplication.
• The subset of unit norm octonions (forming a 7-sphere in O) is closed under multiplication and therefore forms a Moufang loop.
• The basis octonions and their additive inverses form a finite Moufang loop of order 16.
• The set of invertible split-octonions forms a nonassociative Moufang loop, as does the set of unit norm split-octonions. More generally, the set of invertible elements in any octonion algebra over a field F forms a Moufang loop, as does the subset of unit norm elements.
• The set of all invertible elements in an alternative ring R forms a Moufang loop called the loop of units in R.
• For any field F let M(F) denote the Moufang loop of unit norm elements in the (unique) split-octonion algebra over F. Let Z denote the center of M(F). If the characteristic of F is 2 then Z = {e}, otherwise Z = {±e}. The Paige loop over F is the loop M*(F) = M(F)/Z. Paige loops are nonassociative simple Moufang loops. All finite nonassociative simple Moufang loops are Paige loops over finite fields. The smallest Paige loop M*(2) has order 120.
• A large class of nonassociative Moufang loops can be constructed as follows. Let G be an arbitrary group. Define a new element u not in G and let M(G,2) = G ∪ (G u). The product in M(G,2) is given by the usual product of elements in G together with

It follows that and . With the above product M(G,2) is a Moufang loop. It is associative if and only if G is abelian.

• The smallest nonassociative Moufang loop is M(S₃,2) which has order 12.
• Richard A. Parker constructed a Moufang loop of order 213, which was used by Conway in his construction of the monster group. Parker's loop has a center of order 2 with elements denoted by 1, −1, and the quotient by the center is an elementary abelian group of order 212, identified with the binary Golay code. The loop is then defined up to isomorphism by the equations

  A2 = (−1)|A|/4

  BA = (−1)|A∩B|/2AB

  A(BC)= (−1)|A∩B∩C|(AB)C

where |A| is the number of elements of the code word A, and so on. For more details see Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England.