Real and Quasi-central Elements of Finite Groups

An element [Formula: see text] of a finite group [Formula: see text] is said to be real in [Formula: see text] if [Formula: see text] for some [Formula: see text] in [Formula: see text], and quasi-central if [Formula: see text] for each [Formula: see text] in [Formula: see text]. When [Formula: see text] is a 2-group, [Formula: see text] and [Formula: see text], we prove that if every real element in [Formula: see text] of order 4 is quasi-central in [Formula: see text], then every element [Formula: see text] in [Formula: see text] of order 4 is real and [Formula: see text] is normal in [Formula: see text]. Further, we show that a group [Formula: see text] with a Sylow [Formula: see text]-subgroup [Formula: see text] is [Formula: see text]-nilpotent if and only if [Formula: see text] is [Formula: see text]-nilpotent and, for all [Formula: see text] in [Formula: see text], one of the following holds: (a) every element in [Formula: see text] of order [Formula: see text] is quasi-central in [Formula: see text], and if [Formula: see text], every real element in [Formula: see text] of order 4 is quasi-central in [Formula: see text]; (b) every element in [Formula: see text] of order [Formula: see text] is quasi-central in [Formula: see text], and if [Formula: see text], [Formula: see text]; (c) [Formula: see text].