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4.76 (1973)

Solved

Let $G$ be a locally finite group containing an element $a$ of prime order such that the centralizer $C_G(a)$ is finite. Is $G$ almost soluble?

Contributor: V. P. Shunkov

Progress

Yes, it is. P. Fong (Osaka J. Math., 13 (1976), 483–489) has shown (mod CFSG) that $G$ is almost locally soluble. Given this, B. Hartley and T. Meixner (Arch. Math., 36 (1981), 211–213) have proved that $G$ is almost locally nilpotent. Therefore $G$ is almost soluble by (J. L. Alperin, Proc. Amer. Math. Soc., 13 (1962), 175–180) and (G. Higman, J. London Math. Soc., 32 (1957), 321–334). Moreover, by (E. I. Khukhro, Math. Notes, 38, no. 5-6 (1985), 867–870; E. I. Khukhro, Math. USSR–Sb., 71, no. 1 (1992), 51–63) then $G$ is almost nilpotent.

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