B. Hartley asked the following problem:
Problem 1. [2, Problem 3.15]
Let F ∼= P SLm(q) and G ∼= P SLn(q) be two finite simple groups such that F ≤ G with q = p^k for some k and C_G(F) = 1. Does it follow that n is bounded in terms of m?

He suggested several versions of the above problem in [2]. In this talk we answer the above problem by constructing a counter example. In particular we prove the following:

Theorem 2. [1] For any odd n and for any p ≥ n with q = p^k
for some k, there is an embedding φn : PSL2(p) → PSLn(q) such that
C_PSLn(q)(φn(PSL2(p))) = 1.
In particular, we prove that n should be bounded by only p, not by m. We will talk also discuss some of the other versions of the problem related to centralizers in simple locally finite groups.

References

[1] K. Ersoy, “Simple groups embedded as subgroups with trivial centralizer”, in preparation.
[2] B. Hartley, “Simple Locally Finite Groups”, in Finite and Locally Finite Groups , Kluwer Academic, Dordrecht, 1995, 1-44.