Some properties of left 2α-Engel subgroups

Abstract

In 1904, Schur proved his famous result which says that if the central factor group of a given group is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In the present paper, for a given automorphism [Formula: see text] of the group [Formula: see text], we introduce the concept of left [Formula: see text]-Engel, [Formula: see text], and [Formula: see text]-Engel commutator, [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].