On the existence of Engel pairs in certain linear groups

Abstract

Let [Formula: see text] be a group and [Formula: see text]. The 2-tuple [Formula: see text] is said to be an [Formula: see text]-Engel pair, [Formula: see text], if [Formula: see text], [Formula: see text] and [Formula: see text]. Let SL[Formula: see text] be the special linear group of degree [Formula: see text] over the field [Formula: see text]. In this paper, we show that given any field [Formula: see text], there is a field extension [Formula: see text] of [Formula: see text] with [Formula: see text] such that SL[Formula: see text] has an [Formula: see text]-Engel pair for some integer [Formula: see text]. We will also show that SL[Formula: see text] has a [Formula: see text]-Engel pair if [Formula: see text] is a field of characteristic [Formula: see text].