On the Axiomatic Rank of the Quasivariety Mp2
Abstract
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.
DOI 10.14258/izvasu(2015)1.2-33
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Коуровская тетрадь (нерешенные проблемы теории групп). - Новосибирск, 1980.
Ольшанский А.Ю. Условные тождества в конечных групах // Сибирский математический журнал. - 1974. - Т. 15, №6.
Румянцев А.К. О квазитождествах конечных групп // Алгебра и логика. - 1980. - Т. 19, №4.
Будкин А.И. О квазитождествах в свободной группе // Алгебра и логика. - 1976. - Т. 15, №1.
Будкин А.И. Квазитождества нильплтентных групп и групп с одним определяющим соотношением // Алгебра и логика. - 1979. - Т. 18, №2.
Будкин А.И. Аксиоматический ранг квазимногообразия, содержащего свободную разрешимую группу // Математический сборник. - 1980. - Т. 112, №4.
Половникова Е.С. Об аксиоматическом ранге квазимногообразий // Сибирский математический журнал. - 1999. - Т. 40, № 1.
Лодейщикова В.В. О классах Леви, порожденных нильпотентными группами // Сибирский математический журнал. - 2010. - Т. 51, №6.
Лодейщикова В.В. О квазимногообразиях Леви экспоненты 8 // Известия Алт. гос. ун-та. - 2010. - Т. 65, №1/2.
Лодейщикова В.В. О квазимногообразиях Леви экспоненты ps // Алгебра и логика. - 2011. - Т. 50, №1.
Каргаполов М.И., Мерзляков Ю.И. Основы теории групп. - М., 1972.
Будкин А.И. Квазимногообразия групп. - Барнаул, 2002.
Горбунов В.А. Алгебраическая теория квазимногообразий групп. - Новосибирск, 1999.
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