Geodesic
On the orders of free groups of locally finite varieties
Izvestiya. Mathematics, Tome 7 (1973) no. 1, pp. 85-90 
A. Yu. Ol'shanskii. On the orders of free groups of locally finite varieties. Izvestiya. Mathematics, Tome 7 (1973) no. 1, pp. 85-90. http://geodesic.mathdoc.fr/item/IM2_1973_7_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The results of [4] are used here to show that there exists a continuum of locally finite varieties of groups with pairwise distinct “ordinal” functions, thus disproving a conjecture of G. Higman. On the other hand, an example is given of distinct varieties having the same “ordinal” function.

[1] Neiman X., Mnogoobraziya grupp, Mir, M., 1969 | MR

[2] Higman G., “The orders of relatively free groups”, Proc. Internat. Sonf. Theory of Groups (Canberra, 1965), Gordon and Breach, 1967, 153–165 | MR | Zbl

[3] Kovacs L. G., “On the number of varieties of groups”, J. Austral. Math. Soc., 8:3 (1968), 444–446 | DOI | MR | Zbl

[4] Olshanskii A. Yu., “O probleme konechnogo bazisa tozhdestv v gruppakh”, Izv. AN SSSR. Ser. matem., 34 (1970), 376–384

[5] Vaughan-Lee M. R., “Uncountably many varieties of groups”, Bull. London Math. Soc., 2:6 (1970), 280–286 | DOI | MR | Zbl