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.

Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$-group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$. We prove that if $G$ is an ${\mathcal{N}}$-group with ${\it\nu}(G)>\frac{1}{12}$, then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$-groups with ${\it\nu}(G)=\frac{1}{12}$.

The subgroup commutativity degree of a group $G$ is the probability that two subgroups of $G$ commute, or equivalently that the product of two subgroups is again a subgroup. For the dihedral, quasi-dihedral and generalised quaternion groups (all of 2-power cardinality), the subgroup commutativity degree tends to 0 as the size of the group tends to infinity. This also holds for the family of projective special linear groups over fields of even characteristic and for the family of the simple Suzuki groups. In this short note, we show that the family of finite $P$-groups also has this property.

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We also prove this fact.

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for such algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$, where membership of a matrix in $N$ depends only on its ‘invertible part’. The method is based on the availability of estimates for proportions of certain non-singular matrices depending on $N$, so that existing estimation techniques for non-singular matrices can be used to deal with families containing singular matrices. As an application, we investigate primary cyclic matrices, which are used in the Holt–Rees MEATAXE algorithm for testing irreducibility of matrix algebras.

Let ${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants $c,C>0$ such that if $p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.) ${\rm\Gamma}(n,p)$ is free, and if $p\geqslant C\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants $C^{\prime },c^{\prime }>0$ such that if $C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that $x$ and $y$ commute in a compact group’, Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly selected elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probability $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. The situation is more complicated whenever $n,m\gt 1$. If $G$ is a compact Lie group and if its identity component $G_0$ is abelian, then it follows readily that $d_{m,n}(G)$ is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group $G$: for any nonopen closed subgroup $H$ of $G$, the sets $\{g\in G: g^k\in H\}$ for both $k=m$ and $k=n$ have Haar measure $0$. Indeed, we show that if a compact group $G$ satisfies this condition and if $d_{m,n}(G)\gt 0$, then the identity component of $G$is abelian.

We show that if G is a group and A⊂G is a finite set with ∣A2∣≤K∣A∣, then there is a symmetric neighbourhood of the identity S such that Sk⊂A2A−2 and ∣S∣≥exp (−KO(k))∣A∣.

We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.