This Habilitation thesis is a collection of former research of K. Ersoy. In Chapter 1, Introduction, an outline of his research program in general is given. • Chapter 2 is the paper of Ersoy, “Infinite Groups with an Anticentral Element”, [Ers12], appeared in Comm. Alg. 2012. • Chapter 3 is the paper of Ersoy, “Finite Groups with a Splitting Automor- phism of Odd Order”, [Ers16], appeared in Arch. Math. 2016. • Chapter 4 is the paper of Ersoy together with C.K. Gupta, with name “Locally Finite Groups with Centralizers of Finite Rank”, [EG], appeared in Comm. Alg. 2016. • Chapter 5 is the paper of Ersoy together with M. Kuzucuoğlu and P. Shumy- atsky, with name “Locally Finite Groups and Their Subgroups with Small Centralizers”, [EKS], appeared in J. Algebra, 2017. • Chapter 6 is the paper of Ersoy, “Centralizers of p-subgroups in Simple Locally Finite Groups”, [Ers19], that appeared in Glasgow Mathematical Journal, 2020. • Chapter 7 is the paper of Ersoy, together with A. Tortora and M. Tota, with name “On groups with all subgroups subnormal or soluble of bounded derived length”, [ETT], appeared in Glasgow Math. J., 2014
