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4.46 (1973)

Partially Solved

We call a variety of groups a limit variety if it cannot be defined by finitely many laws, while each of its proper subvarieties has a finite basis of identities. It follows from Zorn’s lemma that every variety that has no finite basis of identities contains a limit subvariety.
$\qquad$ a) Give explicitly (by means of identities or by a generating group) at least one limit variety.
$\qquad$ b) Is the set of limit varieties countable?

Contributor: A. Yu. Olshanskii

Progress

b) No, there are continuum of such varieties (P. A. Kozhevnikov, On varieties of groups of large odd exponent, Dep. 1612-V00, VINITI, Moscow, 2000 (Russian); S. V. Ivanov, A. M. Storozhev, Contemp. Math., 360 (2004), 55–62).

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