Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx1x2...xn=xσ(1)xσ(2)...xσ(n), for every (x1,x2,...,xn)∈Sn. A semigroupS is called(r,t)-commutative, wherer,t are in ℕ*, ifx1...xrxr+1...xr+t=xr+1...xr+tx1...xr, for every (x1,x2,...,xr+t∈Sr+t. According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1.