AbstractGiven a finite group G, the invariably generating graph of G is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of G, and two classes are adjacent if and only if they invariably generate G. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most 6.

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Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma $ be a finitely generated Zariski dense subgroup of $G(K)$, and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if $G/\textrm {R}_{u}(G)$ is either trivial or semisimple and $Z$ satisfies certain necessary conditions, then a long random walk on a Cayley graph of $\Gamma $ hits elements of $Z$ with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where $K$ is a global function field.

Let G be a finite simple group. We look for small subsets A of G with the property that, if y ∈ G is chosen uniformly at random, then with good probability y invariably generates G together with some element of A. We prove various results in this direction, both positive and negative.As a corollary of one of these results, we prove that two randomly chosen elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero.Our method is based on the positive solution of the Boston-Shalev conjecture by Fulman and Guralnick, as well as on certain connections between the properties of invariable generation of a group of Lie type and the structure of its Weyl group.

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$ . We prove that either $k(G)< n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$ , or $G$ belongs to explicit families of examples.