Groups with a centralizer of sixth order

Let G be a finite fusion-simple group with a self-centralizing subgroup A of sixth order. It is proved that if the centralizer of the involution from A is an unsolvable subgroup of G of an odd index, then G is isomorphic with the Janko group J₁.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 153–159, July, 1977.The author thanks A. I. Starostin for the formulation of the problem and guidance.