A loop in which the following (equivalent) identities hold:
$$x(y\cdot xz)=(xy\cdot x)z,$$
$$(zx\cdot y)x=z(x\cdot yx),$$
$$xy\cdot zx=x(yz\cdot x).$$
These loops were introduced and studied by R. Moufang . In particular, she proved the following theorem, showing that the loops of this class are close to groups: If the elements $a$, $b$ and $c$ of a Moufang loop satisfy the associativity relation $ab\cdot c=a\cdot bc$, then they generate an associative subloop, that is, a group (Moufang's theorem). A corollary of this theorem is the di-associativity of a Moufang loop: Any two elements of the loop generate an associative subloop.
For commutative Moufang loops, which are defined by the single identity
$$x^2\cdot yz=xy\cdot xz,$$
the following theorem holds: Every commutative Moufang loop with $n$ generators is centrally nilpotent with nilpotency class not exceeding $n-1$ (see ). Central nilpotency is defined analogously to nilpotency in groups (cf. Nilpotent group).
If a loop is isotopic (cf. Isogeny) to a Moufang loop, then it is itself a Moufang loop, that is, the property of being a Moufang loop is universal. Moreover, isotopic commutative Moufang loops are isomorphic.
|||R. Moufang, "Zur Struktur von Alternativkörpern" Math. Ann. , 110 (1935) pp. 416–430|
|||R.H. Bruck, "A survey of binary systems" , Springer (1958)|
Moufang loop. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Moufang_loop&oldid=32766