Moufang loop

2020 Mathematics Subject Classification: Primary: 20N05 [MSN][ZBL]

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 [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 [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

[1]	R. Moufang, "Zur Struktur von Alternativkörpern" Math. Ann. , 110 (1935) pp. 416–430
[2]	R.H. Bruck, "A survey of binary systems" , Springer (1958)