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12.11 (1992)

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Suppose that $G, H$ are countable or finite groups and $A$ is a proper subgroup of $G$ and $H$ containing no non-trivial subgroup normal in both $G$ and $H$. Can $G * _A H$ be embedded in $\operatorname{Sym}(\mathbb{N})$ so that the image is highly transitive?

Contributor: A. M. W. Glass

Progress

Comment of 2013: for partial results, see (S. V. Gunhouse, Highly transitive representations of free products on the natural numbers, Ph. D. Thesis Bowling Green State Univ., 1993; K. K. Hickin, J. London Math. Soc. (2), 46, no. 1 (1992), 81–91).

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