In the first part of this talk we investigate the minimum length of elements in the minimum ideal  of a finite semigroup. We denote this parameter by N(S,A), where A is a generating set of the finite semigroup S, and we call it A-depth of S.  Then we introduce depth parameters for a finite semigroup. We estimate the depth parameters for some families of finite semigroups and give an upper bound for N(S) where S is a wreath product or a direct product of two finite (transformation) monoids.

 In the second part of this talk we are interested in the diameter of a direct power of a finite group. We present the two following conjectures.

Conjecture (strong). Let G be a non trivial finite group. Then the diameter D(G^n) is at most n(|G|-\rank(G)).

Conjecture (weak). Let G be a non trivial finite group. Then there exists a generating set A for G^n of minimum size such that \diam(G^n,A) is at most  n(|G|-\rank(G)).

We show that Abelian groups satisfy the strong conjecture and the weak conjecture is true for nilpotent groups, symmetric groups and the alternating group A_4. We show that the weak conjecture is true for dihedral groups under some restrictions on n. Finally, we present some polynomial upper bounds for the diameter of direct powers of solvable groups.

Keywords. semigroups, generating sets, minimum ideal, diameter of a group, $A$-depth of a semigroup