Moufang loop

In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by . Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra.
Definition.
A Moufang loop is a loop "Q" that satisfies the four following identities for all "x", "y", "z" in "Q" (the binary operation in "Q" is denoted by juxtaposition):
These identities are known as Moufang identities.
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: ("xy")"z" = "x"("yz") then they generate an associative subloop; that is, a group. A corollary of this is that all Moufang loops are "di-associative" (i.e. the subloop generated by any two elements of a Moufang loop is associative and therefore a group). In particular, Moufang loops are power associative, so that exponents "x""n" 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
Left and right multiplication.
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 formula_9 in formula_10. Here formula_11 is bimultiplication by formula_12. The third Moufang identity is therefore equivalent to the statement that the triple formula_13 is an autotopy of formula_10 for all formula_12 in formula_10.
Inverse properties.
All Moufang loops have the inverse property, which means that each element "x" has a two-sided inverse "x"−1 which satisfies the identities:
for all "x" and "y". It follows that formula_18 and formula_19 if and only if formula_20.
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 of finite 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 (Kunen 1996).
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 (or more generally a quasigroup) is the set of formula_23 such that formula_24, formula_25 and formula_26 hold for all formula_27 in the loop.