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11.112 (1990)

Partially Solved

Let $L = L(K(p))$ be the associated Lie ring of a free countably generated group $K(p)$ of the Kostrikin variety of locally finite groups of a given prime exponent $p$. Is it true that
$\qquad$ a) $L$ is a relatively free Lie ring?
$\qquad$ b) all identities of $L$ follow from multilinear identities of $L$?
$\qquad$ c) all identities of $L$ follow from a finite number of identities of $L$?

Contributor: E. I. Khukhro

Progress

b) No, there are two relations of weight 29 which hold in $L(K(7))$ (in two generators, of multiweights (14, 15) and (15, 14)) that are not consequences of multilinear relations (E. O’Brien, M. R. Vaughan-Lee, Int. J. Algebra Comput., 12 (2002), 575–592; M. F. Newman, M. R. Vaughan-Lee, Electron. Res. Announc. Amer. Math. Soc., 4, no. 1 (1998), 1–3).

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