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9.50 (1984)

Solved

Is every 4-Engel group
$\qquad$ a) without elements of order 2 and 5 necessarily soluble?
$\qquad$ b) satisfying the identity $x^5 = 1$ necessarily locally finite?
$\qquad$ c) (R. I. Grigorchuk) satisfying the identity $x^8 = 1$ necessarily locally finite?

Contributor: Yu. P. Razmyslov

Progress

a) Yes; this follows from the local nilpotency of 4-Engel groups (G. Traustason, Int. J. Algebra Comput., 15 (2005), 309–316; G. Havas, M. R. Vaughan-Lee, Int. J. Algebra Comput., 15 (2005), 649–682) and the affirmative answer for locally nilpotent groups in (A. Abdollahi, G. Traustason, Proc. Amer. Math. Soc., 130 (2002), 2827–2836).
b) Yes, it is (M. R. Vaughan-Lee, Proc. London Math. Soc. (3), 74 (1997), 306–334).
c) Yes; moreover, every 4-Engel 2-group is locally finite (G. Traustason, J. Algebra, 178, no. 2 (1995), 414–429).

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