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.