Moufang loop
Encyclopedia
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 Ruth Moufang
.
(the binary operation in Q is denoted by juxtaposition):
for all x, y, z in Q. These identities are known as Moufang identities.
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 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
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 
.
x−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:
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 in 2003: Every finite Moufang loop does have the Lagrange property.
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).
Recall that the nucleus of a loop (or more generally a quasigroup) is the set of x such that
, 
and 
hold for all 
in the loop.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Moufang loop is a special kind of algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
. It is similar to a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang
Ruth Moufang
Ruth Moufang was a German mathematician.Born to a German chemist Dr. Eduard Moufang and Else Fecht Moufang, she studied mathematics at the University of Frankfurt. In 1931 she received her Ph.D. on projective geometry under the direction of Max Dehn, and in 1932 spent a fellowship year in Rome...
.
Definition
A Moufang loop is a loop Q that satisfies the following equivalent identitiesIdentity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
(the binary operation in Q is denoted by juxtaposition):
- z(x(zy)) = ((zx)z)y
- x(z(yz)) = ((xz)y)z
- (zx)(yz) = (z(xy))z
- (zx)(yz) = z((xy)z)
for all x, y, z in Q. These identities are known as Moufang identities.
Examples
- Any groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is an associative loop and therefore a Moufang loop. - The nonzero octonionOctonionIn mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s 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-octonionSplit-octonionIn mathematics, the split-octonions are a nonassociative extension of the quaternions . They differ from the octonions in the signature of quadratic form: the split-octonions have a split-signature whereas the octonions have a positive-definite signature .The split-octonions form the unique split...
s forms a nonassociative Moufang loop, as does the set of unit norm split-octonions. More generally, the set of invertible elements in any octonion algebraOctonion algebraIn mathematics, an octonion algebra over a field F is an algebraic structure which is an 8-dimensional composition algebra over F. In other words, it is a unital nonassociative algebra A over F with a nondegenerate quadratic form N such thatN = NNfor all x and y in A.The most well-known example of...
over a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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 characteristicCharacteristic (algebra)In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
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 fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s. 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 ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
G is abelian.- The smallest nonassociative Moufang loop is M(S3,2) which has order 12.
- Richard A. ParkerRichard A. ParkerRichard A. Parker is a mathematician and freelance computer programmer in Cambridge, England. He invented many of the algorithms for computing the modular character tables of finite simple groups...
constructed a Moufang loop of order 213, which was used by Conway in his construction of the monster groupMonster groupIn the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
. 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 codeBinary Golay codeIn mathematics and electronics engineering, a binary Golay code is a type of error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics....
. 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.
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
- x(xy) = (xx)y left alternativeAlternativityIn abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if y = x for all x and y in G and right alternative if y = x for all x and y in G...
identity - (xy)y = x(yy) right alternativeAlternativityIn abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if y = x for all x and y in G and right alternative if y = x for all x and y in G...
identity - x(yx) = (xy)x flexible identity.
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 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
- z(x(zy)) = (zxz)y
- ((xz)y)z = x(zyz)
- (zx)(yz) = z(xy)z.
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
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 inverseInverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
x−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 L. Lagrange's theoremLagrange's theorem (group theory)
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
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 in 2003: Every finite Moufang loop does have the Lagrange property.
Moufang quasigroups
Any quasigroupQuasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
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:
- Let a be any element of Q, and let e be the unique element such that ae = a. Then for any x in Q, (xa)x = (x(ae))x = (xa)(ex). Cancelling gives x = ex so that e is a left identity element. Now let f be the element such that fe = e. Then (yf)e = (e(yf))e = (ey)(fe) = (ey)e = ye. Cancelling gives yf = y, so f is a right identity element. Lastly, e = ef = f, so e is a two-sided identity element.
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 x such that
, and hold for all in the loop.- See also: Problems in loop theory and quasigroup theoryProblems in loop theory and quasigroup theoryIn mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings...
External links
- LOOPS package for GAP This package has a library containing all nonassociative Moufang loops of orders up to and including 81.