The subset of unit norm octonions is closed under multiplication and therefore forms a Moufang loop.
The subset of unit norm integral octonions is a finite Moufang loop of order 240.
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 octonionalgebra over a fieldF forms a Moufang loop, as does the subset of unit norm elements.
The set of all invertible elements in an alternative ringR forms a Moufang loop called the loop of units in R.
For any fieldF let M denote the Moufang loop of unit norm elements in the split-octonion algebra over F. Let Z denote the center of M. If the characteristic of F is 2 then Z =, otherwise Z =. The Paige loop over F is the loop M* = M/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* 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 ∪. The product in M is given by the usual product of elements in G together with
:
:
:
The smallest nonassociative Moufang loop is M 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 = |A|/4
:BA = |A∩B|/2AB
:A= |A∩B∩C|C
Properties
Associativity
Moufang loops differ from groups in that they need not be associative. A Moufang loop that is associative is a group. The Moufang identities may be viewed as weaker forms of associativity. By setting various elements to the identity, the Moufang identities imply
Moufang's theorem states that when three elements x, y, and z in a Moufang loop obey the associative law: z = x then they generate an associative subloop; that is, a group. A corollary of this is that all Moufang loops are di-associative. In particular, Moufang loops are power associative, so that exponents xn are well-defined. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements. For example, the Moufang identities may be written unambiguously as
The Moufang identities can be written in terms of the left and right multiplication operators on Q. The first two identities state that while the third identity says for all in. Here is bimultiplication by. The third Moufang identity is therefore equivalent to the statement that the triple is an autotopy of for all in.
Inverse properties
All Moufang loops have the inverse property, which means that each element x has a two-sided inversex−1 which satisfies the identities: for all x and y. It follows that and if and only if. Moufang loops are universal among inverse property loops; that is, a loop Q is a Moufang loop if and only if every loop isotope of Q has the inverse property. If follows that every loop isotope of a Moufang loop is a Moufang loop. One can use inverses to rewrite the left and right Moufang identities in a more useful form:
Lagrange property
A finite loop Q is said to have the Lagrange property if the order of every subloop of Q divides the order of Q. Lagrange's theorem in group theory states that every finite group has the Lagrange property. It was an open question for many years whether or not finite Moufang loops had Lagrange property. The question was finally resolved by Alexander Grishkov and Andrei Zavarnitsine, and independently by Stephen Gagola III and Jonathan Hall, in 2003: Every finite Moufang loop does have the Lagrange property. More results for the theory offinite groups have been generalized to Moufang loops by Stephen Gagola III in recent years.
Moufang quasigroups
Any quasigroup satisfying one of the Moufang identities must, in fact, have an identity element and therefore be a Moufang loop. We give a proof here for the third identity: The proofs for the first two identities are somewhat more difficult.
Open problems
Phillips' problem is an open problem in the theory presented by J. D. Phillips at Loops '03 in Prague. It asks whether there exists a finite Moufang loop of odd order with a trivial nucleus. Recall that the nucleus of a loop is the set of such that, and hold for all in the loop.
