[For part I see J. Mycielski, Proc. Am. Math. Soc. 100, 237-241 (1987; Zbl 0626.20024).]
If \(w\) is a word in a free group, we say that \(w\) represents an element \(g\) of a group \(G\), if there is a homomorphism \(h\colon F\to G\) such that \(h(w)=g\). If \(A\) is an infinite set, let \(\text{Sym}(A)\) denote the symmetric group of all permutations of \(A\). The authors give an alternative, simpler and self-contained proof of part of the following result of Lyndon and Mycielski: If \(w\) is not a proper power and \(A\) is infinite, then \(w\) can represent any element of \(\text{Sym}(A)\). The case dealt with here is where \(A\) is countable and the element of \(\text{Sym}(A)\) to be represented has either arbitrarily large finite cycles or an infinite cycle. Furthermore, the authors deal with the interesting question of representing several permutations simultaneously. They also shortly discuss a problem of D. Gale [Math. Intell. 15, No. 3, 56-61 (1993)] whether every permutation of a countable set can be expressed as a product of two cycles. We note that this was already shown earlier [cf. M. Droste, Discrete Math. 47, 35-48 (1983; Zbl 0539.20003)].