Difference between revisions of "Moufang loop"
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A [[Loop|loop]] in which the following (equivalent) identities hold: | A [[Loop|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 | + | 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|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 | 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 | + | 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|Nilpotent group]]). |
If a loop is isotopic (cf. [[Isogeny|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. | If a loop is isotopic (cf. [[Isogeny|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. | ||
Revision as of 14:21, 8 August 2014
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. 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.
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) |
Moufang loop. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moufang_loop&oldid=17216