A left 3-Engel element whose normal closure is not nilpotent

Left 3-Engel elements in locally finite 2-groups

“…Computational evidence suggests that the largest 2-quotient of G and Q coincide. In this section we give an example from [22] of a locally finite 2-group G with a left 3-Engel element a such that a G is not nilpotent. The construction is based on a Lie algebra given in [25].…”

“…The natural question arises whether the analogous result holds for left 3-Engel elements. In [10] it is shown that this is not the case by giving an example of a locally finite 2-group with a left 3-Engel element a such that hai G is not nilpotent. Moreover in [4] an example is given, for each odd prime p, of a locally finite p-group containing a left 3-Engel element x where hxi G is not nilpotent.…”

“…In [19] Newell proved that if a is a right 3-Engel element of G, then a ∈ HP(G); in fact he proved the stronger result that a G is nilpotent of class at most 3. A natural question arises whether a G is nilpotent when a is a left 3-Engel element of G. In [22] it was shown that this is not the case by giving an example of a locally finite 2-group with a left 3-Engel element a such that a G is not nilpotent and in [8] this was generalised to an infinite family of examples. In [9] this was extended to include every odd prime.…”

Left 3-Engel Elements in Locally Finite 2-Groups

“…The natural question arises whether the analogous result holds for left 3-Engel elements. In [10] it is shown that this is not the case by giving an example of a locally finite 2-group with a left 3-Engel element a such that a G is not nilpotent. Moreover in [4] an example is given, for each odd prime p, of a locally finite p-group containing a left 3-Engel element x where x G is not nilpotent.…”

Locally finite $p$-groups with a left 3-Engel element whose normal closure is not nilpotent

“…The natural question arises whether the analogous result holds for left 3-Engel elements. In [10] it has been shown that this is not the case by giving an example of a locally finite 2-group with a left 3-Engel element a such that a G is not nilpotent. In this paper we extend this result to include any odd prime.…”