View source for Moufang loop

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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 [[#References|[1]]]. 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 [[#References|[2]]]). Central nilpotency is defined analogously to nilpotency in groups (cf. [[Nilpotent group]]).

If a loop is isotopic (cf. [[Isotopy (in algebra)]]) 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.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  R. Moufang,   "Zur Struktur von Alternativkörpern"  ''Math. Ann.'' , '''110'''  (1935)  pp. 416–430</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R.H. Bruck,   "A survey of binary systems" , Springer  (1958)</TD></TR>
</table>