Generation of alternating groups by pairs of conjugates (Q1821880)

Let \(A_n\) denote the alternating group of degree \(n\). The main result of the paper is the following Theorem 3.05. Almost all conjugacy classes of \(A_n\) contain a pair of generators. (In other words, the proportion of conjugacy classes in \(A_n\) that contain a pair of generators approaches 1 as \(n\to \infty.)\) The main theorem required the proof of the following Theorems 2.04 and 3.04. Let \(C\) be a conjugacy class (in the symmetric group of degree \(n\)) of type \(T=1^{e(1)}2^{e(2)}3^{e(3)}... \). If \(T\) is not the type of an involution, and if the relation \(\sum _{j\geq 1}e(j)\leq n/2\) holds, then \(C\) contains a pair of elements that generate a primitive group. Almost all partitions of n have a summand \(>1\) and relatively prime to the other summands.