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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650501.png" /></td> </tr></table>
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$$x(y\cdot xz)=(xy\cdot x)z,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650502.png" /></td> </tr></table>
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$$(zx\cdot y)x=z(x\cdot yx),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650503.png" /></td> </tr></table>
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$$xy\cdot zx=x(yz\cdot x).$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650506.png" /> of a Moufang loop satisfy the associativity relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650507.png" />, 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.
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650508.png" /></td> </tr></table>
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$$x^2\cdot yz=xy\cdot xz,$$
  
the following theorem holds: Every commutative Moufang loop with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m0650509.png" /> generators is centrally nilpotent with nilpotency class not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065050/m06505010.png" /> (see [[#References|[2]]]). Central nilpotency is defined analogously to nilpotency in groups (cf. [[Nilpotent group|Nilpotent group]]).
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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)
How to Cite This Entry:
Moufang loop. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moufang_loop&oldid=32766
This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article