On profinite groups in which commutators are covered by finitely many subgroups

For a family of group words w we show that if G is a profinite group in which all w-values are contained in a union of finitely many subgroups with a prescribed property, then the verbal subgroup w(G) has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If G contains finitely many subgroups G (1), G (2), . . . , G (s) of finite exponent e whose union contains all gamma (k) -values in G, it is shown that gamma (k) (G) has finite (e, k, s)-bounded exponent. If G contains finitely many subgroups G (1), G (2), . . . , G (s) of finite rank r whose union contains all gamma (k) -values, it is shown that gamma (k) (G) has finite (k, r, s)-bounded rank.