Symmetric generating set of the groups \(A_{2n+1}\) and \(S_{2n+1}\) using \(S_n\) and an element of order two (Q1380968)

The author shows how to generate either the symmetric or alternating group (depending whether \(n\) is odd or even) of degree \(2n+1\) using three specific permutations, the first two of which generate a subgroup isomorphic to \(S_n\) and the third of which has order two. The third generator also commutes with a subgroup isomorphic to \(S_{n-1}\), and \(n\) conjugates of this involution are shown to form a ``symmetric generating set'' for the whole group. The paper refers to the author's unpublished Ph.D. thesis for most of its background and motivation, but the notion of ``symmetric generating set'' used here appears to be that of \textit{R. T. Curtis} [Lond. Math. Soc. Lect. Note Ser. 165, 380-396 (1992; Zbl 0812.20002)].