Some properties of central kernel and central autocommutator subgroups

Abstract

A classical result of Schur states that if the central quotient [Formula: see text] of a group [Formula: see text] is finite, then the commutator subgroup [Formula: see text] is also finite. In this paper we introduce the notion of central autocommutator subgroup of a given group [Formula: see text]. We study this concept and give some new results concerning the central kernel subgroup of [Formula: see text], which was first introduced by F. Haimo in 1955. More precisely, the analogue of Schur’s result is proved. We also construct some upper bounds for the order of central kernel and central autocommutator subgroups of [Formula: see text] in terms of the order of central kernel quotient of [Formula: see text].