Moufang implies diassociative
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two loop properties. That is, it states that every loop satisfying the first loop property (i.e., Moufang loop) must also satisfy the second loop property (i.e., diassociative loop)
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Statement
Any Moufang loop is a diassociative loop.
Related facts
Similar facts
Facts used
- Moufang implies alternative: A Moufang loop is an alternative loop, i.e., it satisfies the left alternative law
and the right alternative law
.
- Moufang's theorem: This states that if
is a Moufang loop, and
are (not necessarily distinct) elements of
such that
, then the subloop of
generated by
is a group, i.e., it is associative.
Proof
Given: A Moufang loop . Elements
(not necessarily distinct).
To prove: The subloop of generated by
is associative.
Proof: Setting , we see that by fact (1),
(using the right alternative law part). Hence, by fact (2), the subloop generated by
is a group. But since
, this is the same as the subloop generated by
and
, completing the proof.