In this paper, we study the probability distribution of the word map $w(x_{1},x_{2},\ldots ,x_{k})=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{k}^{n_{k}}$ in a compact Lie group. We show that the probability distribution can be represented as an infinite series. Moreover, in the case of the Lie group $\text{SU}(2)$, our computations give a nice convergent series for the probability distribution.

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group.

We discuss some questions related to the generation of supersoluble groups. First we prove that the number of elements needed to generate a finite supersoluble group G with good probability can be quite a lot larger than the smallest cardinality d(G) of a generating set of G. Indeed, if G is the free prosupersoluble group of rank d ⩾ 2 and dP(G) is the minimum integer k such that the probability of generating G with k elements is positive, then dP(G) = 2d + 1. In contrast to this, if k – d(G) ⩾ 3, then the distribution of the first component in a k-tuple chosen uniformly in the set of all the k-tuples generating G is not too far from the uniform distribution.

We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

We will compute the subgroup permutability degree of the projective special linear groups PSL(2,pn).