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For a finite group group‎, ‎denote by $\mathcal V(G)$ the smallest positive integer $k$ with the property that the probability of generating $G$ by $k$ randomly chosen elements is at least $1/e.$ Let $G$ be a finite soluble group‎. ‎{Assume} that for every $p\in \pi(G)$ there exists $G_p\leq G$ such that $p$ does not divide $|G:G_p|$ and ${\mathcal V}(G_p)\leq d.$ Then ${\mathcal V}(G)\leq d+7.$‎

We prove that every finite group $G$ contains a three-generated subgroup $H$ with the following property‎: ‎a prime $p$ divides the degree of an irreducible character of $G$ if and only if it divides the degree of an irreducible character of $H.$ There is no analogous result for the prime divisors of the sizes of the conjugacy classes‎.

We analyze some properties of the distribution Q G,k of the first component in a k -tuple chosen uniformly‎ ‎in the set of all the k -tuples generating a finite group G (the limiting distribution of the product replacement algorithm)‎. ‎In particular‎, ‎we concentrate our attention on‎ ‎the study of the variation distance β k (G) between Q G,k and the uniform distribution‎. ‎We review some known results‎, ‎analyze several examples and propose some intriguing open questions‎.

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilisticzeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many non abelian composition factors of $G$ are isomorphic to $PSL(2,p)$ for some prime $p$, then $G$ contains only finitely many maximal subgroups.