The primitive permutation groups of degree less than 2500

In this paper we use the O’Nan–Scott Theorem and Aschbacher’s theorem to classify the primitive permutation groups of degree less than 2500. MSC: 20B15, 20B10 1 Historical Background The classification of the primitive permutation groups of low degree is one of the oldest problems in group theory. The earliest significant progress was made by Jordan, who in 1871 counted the primitive permutation groups of degree d for d ≤ 17 [20], and stated that a transitive group of degree 19 is A19, S19, or a group of affine type. There were various minor omissions in Jordan’s work: degree 9 was corrected by Cole [6, 7], and the remaining degrees up to 17 were corrected by Miller in a long series of papers at the end of the 19th century [35, 36, 37, 38, 39, 40, 41]. In these papers Miller also correctly tabulated the number of soluble primitive groups of degree less than 24. By 1912, the classification up to degree 20 had been completed by [34, 3] for degrees 18 and 20, respectively. After this there was a hiatus: the lists of groups were becoming so long as to be unwieldy for hand calculations, and the chance of an error arising in such an extended calculation was too great. The birth of practical symbolic computation in the 1960s renewed interest in this old problem, and by 1970 Sims had redetermined the list of primitive groups of degree up to 20 [47]. Sims also classified the primitive groups of degree up to 50: this was never published, but the list was widely circulated in manuscript form, and the resulting groups formed one of the earliest databases in computational group theory, becoming part of first CAYLEY [5], and later GAP [13] and Magma [2]. Various people worked on this problem in the 1970s and early 1980s, but the next dramatic leap forward came as a result of the announcement of the Classification of Finite Simple Groups (CFSG), after which Dixon and Mortimer used the O’Nan–Scott Theorem to classify the primitive groups with insoluble socles of degree less than 1000 [10]. ∗The author was partially supported by EPSRC grant number GR/S30580 and partially funded by the Australian Research Council. Part of this work was carried out at the University of Sydney.