Consider a finite semigroup S with a generating set A. By the length of an element s in S, with respect to A, we mean the minimum length of a sequence which represents s in terms of generators in A. Define the parameter N(S, A) to be the minimum length of elements contained in the minimum ideal of S. Let the parameters N(S), M(S) be the minimum and the maximum of N(S, A) over all generating sets of minimum size, respectively; and denote by M’(S) the maximum of N(S, A) over all generating sets of S. In the first part of this talk, we shall present some classes of semigroups for which the above-mentioned parameters have been estimated. Furthermore, we will present an upper bound for N(S), provided that S is a wreath product of two finite semigroups. If the factors of the product do not have trivial groups of units, the diameter of a semidirect product of groups will appear in the obtained upper bound; and in the special case that one of factors has trivial group of units, the diameter of a direct power of a finite group will appear. In the second part of this talk, we will discuss the diameter of a direct power of a finite group.