Moufang loop

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A loop in which the following (equivalent) identities hold:

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 , and of a Moufang loop satisfy the associativity relation , 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

the following theorem holds: Every commutative Moufang loop with generators is centrally nilpotent with nilpotency class not exceeding (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.


[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. V.D. Belousov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098