Engel Lie-Algebras

On locally finite 𝑝-groups satisfying an Engel condition

“…In [24] M. Vaughan-Lee proved that 4-Engel groups of exponent 5 are locally finite but for primes greater than 5 the question remains open. However, if p ≥ 7, then every locally finite 4-Engel p-group is nilpotent of class at most 7 [22] (see also [9], [21]). …”

The nilpotency class of finite groups of exponent 𝑝

“…Higman's algebra L is also known to be nilpotent for the case n = 3, char(A:) ^ 2, and for the case n = 5, char(A:) ^2, 3 (see [14]). On the other hand Traustason [13] has found Engel-3 Lie algebras of characteristic 2 in which the ideal generated by an element is not nilpotent, and I have similar examples of Engel-5 Lie algebras of characteristic 3.…”

“…Let / be the ideal of L generated by {ab4\a, b £ L} . Traustason [13] has shown that L/I is nilpotent of class 7, which implies that Ls < I. Now let S be a subset of the generators of L, let Ls be the subalgebra of L generated by S, and let Is be the ideal of L generated by {ab4\a, b £ Ls} .…”

“…If L is an Engel-3 Lie algebra over a field K, then L is nilpotent of class at most 4 provided char(A^) ^2, 5 (see [6] and [13]). In the case when char(ZT) = 2, then an m-generator Engel-3 Lie algebra over K is nilpotent of class at most 2(m + l)6 (see [14]), and in the case when char(AT) = 5, then an m-generator Engel-3 Lie algebra over K is nilpotent of class at most 2m .…”

“…In the case when char(ZT) = 2, then an m-generator Engel-3 Lie algebra over K is nilpotent of class at most 2(m + l)6 (see [14]), and in the case when char(AT) = 5, then an m-generator Engel-3 Lie algebra over K is nilpotent of class at most 2m . Engel-4 Lie algebras are nilpotent of class at most 8 provided the characteristic of the ground field is not 2, 3 , or 5 (see [13]). Traustason [14] has shown that an m-generator Engel-4 Lie algebra over a field of characteristic 2 is nilpotent of class at most 2(m + l)6.…”