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10.66 (1986)

Solved

Is a group $G$ non-simple if it contains two non-trivial subgroups $A$ and $B$ such that $AB \neq G$ and $AB^g = B^gA$ for any $g \in G$? This is true if $G$ is finite (O. H. Kegel, Arch. Math., 12, no. 2 (1961), 90–93).

Contributor: A. N. Fomin, V. P. Shunkov

Progress

No; for example, for $G$ being any simple linearly ordered group and $A$ and $B$ arbitrary proper convex subgroups of $G$ (V. V. Bludov, Letter of February, 12, 1997).

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