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14.88 (1999)

Solved

We say that an element $u$ of a group $G$ is a test element if for any endomorphism $\varphi$ of $G$ the equality $\varphi(u) = u$ implies that $\varphi$ is an automorphism of $G$. Does a free soluble group of rank 2 and derived length $d > 2$ have any test elements?

Contributor: B. Fine, V. Shpilrain

Progress

Yes, it does (E. I. Timoshenko, Algebra and Logic, 45, no. 4 (2006), 254–260.)

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