Geodesic
The axiomatic rank of a quasivariety containing a free solvable group
Sbornik. Mathematics, Tome 40 (1981) no. 4, pp. 583-590 
A. I. Budkin. The axiomatic rank of a quasivariety containing a free solvable group. Sbornik. Mathematics, Tome 40 (1981) no. 4, pp. 583-590. http://geodesic.mathdoc.fr/item/SM_1981_40_4_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathfrak P$ be the class of groups for which the commutator subgroup of each subgroup intersects the center of that subgroup in the identity. In this paper it is shown that if a quasivariety $\mathfrak M$ contains a non-Abelian group free in $\mathfrak A^2$ and if $\mathfrak M\subseteq\mathfrak P$, then $\mathfrak M$ cannot be given by a system of quasi-identities in a finite number of variables. Bibliography: 9 titles.

[1] A. I. Budkin, “O kvazitozhdestvakh v svobodnoi gruppe”, Algebra i logika, 15:1 (1976), 39–52 | MR | Zbl

[2] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, izd-vo “Nauka”, Moskva, 1972 | MR

[3] A. I. Maltsev, Algebraicheskie sistemy, izd-vo “Nauka”, Moskva, 1970 | MR

[4] A. I. Maltsev, “O vklyuchenii assotsiativnykh sistem v gruppy. I”, Matem. sb., 6(48) (1939), 331–336 | Zbl

[5] A. I. Maltsev, “O vklyuchenii assotsiativnykh sistem v gruppy. II”, Matem. sb., 8(50) (1940), 251–263

[6] A. I. Maltsev, “O svobodnykh razreshimykh gruppakh”, DAN SSSR, 130:3 (1960), 495–498

[7] X. Neiman, Mnogoobraziya grupp, izd-vo “Mir”, Moskva, 1969 | MR

[8] V. N. Remeslennikov, “Primer gruppy, konechno opredelennoi v mnogoobrazii $\mathfrak A^5$, s nerazreshimoi problemoi ravenstva”, Algebra i logika, 12:5 (1973), 577–602 | MR | Zbl

[9] A. Yu. Olshanskii, “Uslovnye tozhdestva v konechnykh gruppakh”, Sib. matem. zh., 15:2 (1974), 1409–1413