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

Abstract

For any odd prime [Formula: see text], we give an example of a locally finite [Formula: see text]-group [Formula: see text] containing a left 3-Engel element [Formula: see text] where [Formula: see text] is not nilpotent.

Left 3-Engel elements in locally finite 2-groups

“…Remark. We used the Magma implementation of the nilpotent quotient algorithm to determine that the largest nilpotent quotient Q of G has order 2 71 and class 10. If we impose the two relations of Theorem 3.2, then Q has order 2 38 and class 9.…”

“…In Section 4, following [8,22], we gave an example of a locally finite 2-group G with a left 3-Engel element a such that a G is not nilpotent. We now provide such an example for locally finite p-groups where p is any odd prime [9]. The odd case is more involved and the construction quite different from the p = 2 case.…”

“…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.…”

“…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. Thus for each prime p there is a locally finite p-group G containing a left 3-Engel element a such that a G is not nilpotent.…”

Left 3-Engel Elements in Locally Finite 2-Groups

“…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.…”