We show that there exists a non-linear, locally finite, simple group such that the centralizer of every non-trivial element is (locally solvable)-by-finite.
MEIERFRANKENFELD, U. (2007) "A Non-Linear Locally Finite Simple Group with a p-Group as Centralizer," Turkish Journal of Mathematics: Vol. 31: No. 5, Article 7. Available at: https://journals.tubitak.gov.tr/math/vol31/iss5/7