Abstract
This chapter discusses a characterization of PSp4 (3m) by the centralizer of an element of order three. The proof uses a characterization of L2(32n) as a Cθθ-group, which is stated below as a conjecture. Much work has been done in this area, and recent results suggest that the following conjecture is true. If G be a finite group which satisfies the following: (1) G contains an elementary abelian 3-group D of order q = 32n with CG (d) = D for all d ∈ D#. (2) G has at least 2 classes of 3-elements. (3) D is not normal in G. Then, [NG (D):D] = q – 1/2 and G ≅ L2(q).
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