On locally finite Cpp-groups

A group G is called a Cpp-group for a prime number p, if G has elements of order p and the centralizer of every non-trivial p-element of G is a p-group. In this paper we prove that the only infinite locally finite simple groups that are Cpp-groups are isomorphic either to PSL(2, K) or, if p = 2, to Sz(K), with K a suitable algebraic field over GF(p). Using this fact, we also give some structure theorems for infinite locally finite Cpp-groups.