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6.63 (1978)

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An infinite group $G$ is called a monster of the first kind if it has elements of order $> 2$ and for any such an element $a$ and for any proper subgroup $H$ of $G$, there is an element $g$ in $G \setminus H$, such that $\langle a, a^g \rangle = G$. Classify the monsters of the first kind all of whose proper subgroups are finite.

Contributor: V. P. Shunkov

Progress

The centre of such a group coincides with the set of elements of order $\leqslant 2$ (V. P. Shunkov, Algebra and Logic, 7 (1968), no. 1 (1970), 66–69). An infinite group all of whose proper subgroups are finite is a monster of the first kind if its centre coincides with the set of elements of order $\leqslant 2$ (A. I. Sozutov, Algebra and Logic, 36, no. 5 (1997), 336–348). There are continuously many such groups (A. Yu. Olshanskii, Geometry of defining relations in groups, Kluwer, Dordrecht, 1991).

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