Prime power order implies nilpotent for Moufang loops

ANALOGY: This is an analogue in Moufang loops of a fact encountered in group. The old fact is: prime power order implies nilpotent.
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Statement

Suppose $L$ is a Moufang loop of prime power order, i.e., a Moufang loop whose underlying set has order a prime power. Then, $L$ is a nilpotent Moufang loop (i.e., a finite nilpotent Moufang loop).

Related facts

Similar facts

• Prime power order implies nilpotent (in groups)

Opposite facts

• Prime power order not implies nilpotent for Lie rings