In my paper [1], a normalization factor of |G|−1 was missing in the statements of Theorem 2.4, Theorem 2.6 and Corollary 2.7.

The generating graph $\unicode[STIX]{x1D6E4}(H)$ of a finite group $H$ is the graph defined on the elements of $H$, with an edge between two vertices if and only if they generate $H$. We show that if $H$ is a sufficiently large simple group with $\unicode[STIX]{x1D6E4}(G)\cong \unicode[STIX]{x1D6E4}(H)$ for a finite group $G$, then $G\cong H$. We also prove that the generating graph of a symmetric group determines the group.

We study the distribution of products of conjugacy classes in finite simple groups, obtaining effective two-step mixing results, which give rise to an approximation to a conjecture of Thompson.

Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL(2, q) to all (non-abelian) finite simple groups.

In particular it follows that, if G is a finite simple group, and A, B ⊆ Gt for t ⩾ 2 are subsets of fixed positive densities, then, as a = (a1, . . ., at) ∈ A and b = (b1, . . ., bt) ∈ B are chosen uniformly, the interleaved product a • b:=a1b1 . . . atbt is almost uniform on G (with quantitative estimates) with respect to the ℓ∞-norm.

It also follows that the communication complexity of an old decision problem related to interleaved products of a, b ∈ Gt is at least Ω(t log |G|) when G is a finite simple group of Lie type of bounded rank, and at least Ω(t log log |G|) when G is any finite simple group. Both these bounds are best possible.

We show that the proportion of permutations $g$ in $S_{\!n}$ or $A_{n}$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^{{\it\varepsilon}}\rceil$ is at least a constant multiple of ${\it\varepsilon}$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^{{\it\varepsilon}}\rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2\rfloor ^{{\it\varepsilon}}\rceil$ for a symplectic or orthogonal group.

We give a new elementary construction of Ree's family of finite simple groups of type 2G2, which avoids the need for the machinery of Lie algebras and algebraic groups. We prove that the groups we construct are simple of order q3(q3 + 1)(q − 1) and act doubly transitively on an explicit set of q3 + 1 points, where q = 32k+1. Moreover, our construction is practical in the sense that generators for the groups and many of their maximal subgroups may easily be obtained.