Locally nilpotent 4-Engel groups are Fitting groups

Abstract

Let G be a locally nilpotent 4-Engel group. We show that the normal closure of any element from G is nilpotent of class at most 4. When G has no element of order 2 or 5, the normal closure has class at most 3. These bounds are sharp.

Computing with 4-Engel groups

“…Corollary 2 of [6] states that if G is an m-generator, 4-Engel group then G is nilpotent of class at most 4m, and if G has no elements of order 2, 3 or 5 then G is nilpotent of class at most 7. Given our result that 4-Engel groups are locally nilpotent, this corollary follows from Traustason's work [9,10] on the class of locally nilpotent 4-Engel groups. But if we restrict our attention to p-groups then we can sharpen this corollary.…”

On a problem of P. Hall for Engel words

“…In case of 3-Engel groups there are some classic results in [6,8] which allow us to conclude that every d-generated 3-Engel group is nilpotent of class at most 2d. Similarly for 4-Engel groups, [5,15] allow us to conclude that every d-generated 4-Engel group is nilpotent of class at most 4d. Therefore…”

On a Problem of P. Hall for Engel Words Ii

“…(vi) The lower bound follows from the definitions. Applying again the results in [10,22], we may argue as in (iv) and find the required upper bound.…”

“…For the lower bound, the arguments of (i) and (iii) imply that G/E 4 (G) is a 4-Engel group. From [10,22], we know that G/E 4 (G) is nilpotent of class at most 4h. Then γ 4h+1 (G) ≤ E 4 (G).…”