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4.33 (1973)

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Let $\mathfrak{K}_n$ be the class of all groups with a single defining relation in the variety of soluble groups of derived length $n$.
$\qquad$ a) Under what conditions does a $\mathfrak{K}_n$-group have non-trivial center? Can a $\mathfrak{K}_n$-group, $n \geqslant 2$, that cannot be generated by two elements have non-trivial centre?
$\qquad$ b) Describe the abelian subgroups of $\mathfrak{K}_n$-groups.
$\qquad$ c) Investigate the periodic subgroups of $\mathfrak{K}_n$-groups.

Contributor: M. I. Kargapolov

Progress

Editors’ comment (1998): These questions were answered for $n = 2$ (E. I. Timoshenko, Siberian Math. J., 14, no. 6 (1973), 954–957; Math. Notes, 64, no. 6 (1998), 798–803).

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