A question of Higman (Q1814620)

If a finitely generated group \(G\) is embeddable into \(\hbox{Aut}_ r(\omega)\), the group of recursive permutations of \(\omega\), then the word problem for \(G\) is obviously coenumerable. G. Higman asked whether the converse was true. The author answers the question in the negative. He shows that the group \(G_ g\) generated by any nonrecursive permutation \(g\) of \(\omega\) and the permutations \((01), (01)(23)\dots\) and \((12)(34)\dots\) can not be embedded into \(\hbox{Aut}_ r(\omega)\) and constructs a nonrecursive \(g\) such that the word problem for \(G_ g\) is coenumerable.