
12(85) 2015 MATHEMATICS
S.A. Shakhova
On the Axiomatic Rank of the Quasivariety M^{p2}
Let ρ be a prime number, ρ ≠ 2, H_{p2} be a group with the following presentation in the variety of nilpotent groups of class at most two: H_{p2} = gr(x, yx^{p2} = y^{p2} = [x, y]^{p} = 1) and let qH_{p2} be the quasivariety generated by the group H_{p2} . We denote M^{p2} = L(qH_{p2} ), where L(qH_{p2} ) is the Levi class generated by the quasivariety qH_{p2}. By definition, the Levi class L(qH_{p2} ) is a class of all groups where the normal closure of every there element belongs to qH_{p2} . It is well known that the Levi class generated by a quasivariety is a quasivariety too. Besides, the list of quasiidentities that defines the quasivariety M^{p2} is known. There is an infinite number of quasiidentities with an arbitrarily large number of variables in the list. The base of the quasivariety is the set of quasiidentities 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 M^{p2} is finite? It is proven that the axiomatic rank of the quasivariety M^{p2} is finite, and there is a base of the quasivariety M^{p2} that depends on three variables.
DOI 10.14258/izvasu(2015)1.233
Key words: quasivariety, quasiidentity, group, nilpotent group, Levi class, axiomatic rank
