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15.75 (2002)

Partially Solved

a) Does there exist a sequence of identities in two variables $u_1 = 1$, $u_2 = 1, \dots$ with the following properties: 1) each of these identities implies the next one, and 2) an arbitrary finite group is soluble if and only if it satisfies one of the identities $u_n = 1$?
b) Consider the sequence $u_1 = [x, y], \dots, u_{n+1} = [[u_n, x], [u_n, y]]$. Is it true that an arbitrary finite group is soluble if and only if it satisfies one of these identities $u_n = 1$?

Contributor: B. I. Plotkin

Progress

a) Yes, such sequences exist (T. Bandman, F. Grunewald, G.-M. Greuel, B. Kunyavskii, G. Pfister, E. Plotkin, Compositio Math., 142 (2006), 734–764; J. N. Bray, J. S. Wilson, R. A. Wilson, Bull. Lond. Math. Soc., 37 (2005), 179–186; E. Ribnere, Monatsh. Math., 157 (2009), 387–401).

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