Representing the quotient groups of a finite permutation group (Q1604392)

Let \(G\) be a finite group. The authors define the ``minimal degree'' of \(G\) to be the least integer \(d\) such that \(G\) can be embedded in the symmetric group \(S_d\) (note that this is not the same as the classical meaning of minimal degree of a permutation group). The minimal degree of a quotient of \(G\) may be exponentially larger than the minimal degree of \(G\). Indeed, examples are known of groups \(G\) of minimal degree \(4k\) for which some quotient \(G/N\) has minimal degree \(2^{k+1}\) [see \textit{P. M. Neumann}, in Groups, Proc. Int. Conf., St. Andrews/Scotl. 1985, Lond. Math. Soc. Lect. Note Ser. 121, 59-92 (1986; Zbl 0612.20001)]. On the other hand, a result of \textit{L. G. Kovács} and \textit{C. E. Praeger} [Bull. Aust. Math. Soc. 62, No. 2, 311-317 (2000; Zbl 0973.20001)] shows that, if \(G/N\) has no nontrivial normal Abelian subgroup, then the minimal degree of \(G/N\) is no larger than the minimal degree of \(G\). The main results of the present paper are as follows. If \(G\leq S_d\) and \(c=4.5\), then \(G\) has a subgroup of index at most \(c^{d-1}/d\) all of whose composition factors are alternating groups (it is further shown that these alternating groups act ``\(c\)-naturally'' where the latter is a technical condition too complicated to explain here). Using this theorem and the result of Kovács and Praeger quoted above, the authors prove: If a group \(G\) has minimal degree \(d\), then every quotient \(G/N\) has minimal degree at most \(c^{d-1}\).