Finite groups in which the centralizers of elements of order three are nilpotent

In this paper it is proved that a finite group of 3-rank 1, in which the soluble radical is trivial and the centralizers of elements of order 3 are nilpotent, is isomorphic to one of the following groups: $L_3(4)$, $L_3^*(4)$, $PGL(2, 3^n)$ or $H(3^n)$, $n\geqslant2$.
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