1.2 (1965)

Let $G$ be a group, $F$ a free group with free generators $x_1, \dots, x_n$, and $R$ the free product of $G$ and $F$. An equation (in the unknowns $x_1, \dots, x_n$) over $G$ is an expression of the form $v(x_1, \dots, x_n) = 1$, where on the left is an element of $R$ not conjugate in $R$ to any element of $G$. We call $G$ algebraically closed if every equation over $G$ has a solution in $G$.

Do there exist algebraically closed groups?