On three-Engel groups

Abstract

The following conditions for a group G are investigated:(i) maximal class n subgroups are normal, (ii) normal closures of elements have nilpotency class n at most, (iii) normal closures are n-Engel groups, (iv) G is an (n+l)-Engel group. Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1 . Here the question is settled for n = 2 as follows: conditions ( i i ) , (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.

On locally finite 𝑝-groups satisfying an Engel condition

“…Whereas 3-Engel groups are now quite well understood (see for example [2], [10], [12], [13], [15]), relatively little is known about the structure of 4-Engel groups. In particular it is still unknown whether every 4-Engel p-group needs to be locally finite.…”

Subnormality conditions in non-torsion groups

“…Conversely, by [6] we have for any group G the conditions G being 3-Engel and the normal closure The result for n = 2, corresponding to the above corollary, is due to Heineken [5]. The result of the next corollary appears in [3] for n < 5 and for n ^ 6, provided the group contains no elements of order ^ n -1.…”

On Nonabelian Tensor Analogues of 2-Engel Conditions

“…Again we may assume that |G : N i | < ∞ and by Theorem 4 we also have |G : R x −1 , hence G is 3 ⊗ -Engel. In view of [10] it is likely that a 3 ⊗ -Engel group G with |G : R ⊗ 2 (G)| < ∞ has a finite normal covering by 2 ⊗ -Engel subgroups, but we have not been able to (dis)prove this, since there are no known tensor analogues of results regarding 3-Engel groups [12].…”

“…It is not difficult to see that if G has a finite covering by 2 ⊗ -Engel normal subgroups, then G is 3 ⊗ -Engel and |G : R ⊗ 2 (G)| < ∞. For the reverse conclusion one would probably need the characterization of 3 ⊗ -Engel groups by their normal closures analogous to [12].…”