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On the laws of certain varieties of groups

Let m and n be coprime positive integers. The variety (consisting of all groups G such that for some normal subgroup H of G, H is abelian of exponent dividing m and G/H is abelian of exponent dividing n) and the variety both satisfy the following three laws: all elements have order dividing mn; the commutator of two mth powers has order dividing m; the commutator of two nth powers has order dividing n. It is proved that any law which holds in both these varieties (notably that commutators commute) is a consequence of the above three laws.