A graph is edge-primitive if its automorphism group acts primitively on the edge set, and $2$ -arc-transitive if its automorphism group acts transitively on the set of $2$ -arcs. In this paper, we present a classification for those edge-primitive graphs that are $2$ -arc-transitive and have soluble edge-stabilizers.

A cover of a group is a finite collection of proper subgroups whose union is the whole group. A cover is minimal if no cover of the group has fewer members. It is conjectured that a group with a minimal cover of nilpotent subgroups is soluble. It is shown that a minimal counterexample to this conjecture is almost simple and that none of a range of almost simple groups are counterexamples to the conjecture.

This is a contribution to the study of line-transitive groups of automorphisms of finite linear spaces. Groups which are almost simple are of particular importance. In this paper almost simple line-transitive groups whose socle is an alternating group are classified. It is proved that the only alternating groups to occur are those of degrees 7 and 8, and that only one linear space occurs, namely a well-known space with 15 points and 35 lines. Although much of the proof exploits special properties of alternating groups, some general theory of groups acting line-transitively on finite linear spaces is developed.

Let $G$ be a finite group.It is proved that if the probability that two randomly chosen elements of $G$ generate a soluble group is greater than $\frac{11}{30}$ then $G$ itself is soluble.The bound is sharp, since two elements of the alternating group $A_5$ generate $A_5$ withprobability $\frac{11}{30}$.Similar probabilistic statements are proved concerning nilpotency and the property of having odd order.It is also proved that there is a number $\kappa$,strictly between $0$ and $1$, with the following property. Let $\cal X$ be any class of finite groups which is closed for subgroups, quotient groups and extensions.If the probability that two randomly chosen elementsof $G$ generate a group in $\cal X$ is greater than $\kappa$ then $G$ is in $\cal X$. The proofs use the classification of the finite simple groups and also some of the detailed information now available concerning maximal subgroups of finite almost simple groups. 1991 Mathematics Subject Classification:20F16, 20D06, 20D08, 60B99.