On the Axiomatic Rank of the Quasivariety Mp2
Let ρ be a prime number, ρ ≠ 2, Hp2 be a group with the following presentation in the variety of nilpotent groups of class at most two: Hp2 = gr(x, y||xp2 = yp2 = [x, y]p = 1) and let qHp2 be the quasivariety generated by the group Hp2 . We denote Mp2 = L(qHp2 ), where L(qHp2 ) is the Levi class generated by the quasivariety qHp2. By definition, the Levi class L(qHp2 ) is a class of all groups where the normal closure of every there element belongs to qHp2 . It is well known that the Levi class generated by a quasivariety is a quasivariety too. Besides, the list of quasi-identities that defines the quasivariety Mp2 is known. There is an infinite number of quasi-identities with an arbitrarily large number of variables in the list. The base of the quasivariety is the set of quasi-identities that defines this quasivariety. By the definition, the axiomatic rank of the quasivariety is finite if there is a base of the quasivariety with a finite number of variables. The following question arises: is it true that the axiomatic rank of the quasivariety Mp2 is finite? It is proven that the axiomatic rank of the quasivariety Mp2 is finite, and there is a base of the quasivariety Mp2 that depends on three variables.
Key words: quasivariety, quasi-identity, group, nilpotent group, Levi class, axiomatic rank
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