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10.74 (1986)

Open

Suppose that a group $G$ contains an element $a$ of prime order such that its centralizer $C_G(a)$ is finite and all subgroups $\langle a, a^g \rangle$, $g \in G$, are finite and almost all of them are soluble. Is $G$ locally finite? This problem is closely connected with 6.56.

Contributor: V. P. Shunkov

Progress

The question was solved in the positive for a number of very important partial cases by the author (Abstracts on Group Theory of the Mal’cev Int. Conf. on Algebra, Novosibirsk, 1989, p. 145 (Russian)).

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