Computing Chief Series, Composition Series and Socles in Large Permutation Groups

In this paper, we describe the theory and implementation of algorithms for computing chief series, composition series and socles in large permutation groups. The theory is valid for permutation degrees up to 10 000 000. Most parts of the algorithms have been implemented within the Magma Computational Algebra System (Bosma and Cannon, 1993; Bosma et al., 1994; Bosma et al., 1997; Cannon and Playoust, 1996), and the remainder of them will be shortly. As we shall see from the performance descriptions on some examples in Section 6 below, they are currently practical for degrees up to several hundred thousand for many types of examples. (The precise limits depend mainly on the amount of computer memory available. For the performance figures below, we had 256 Mb.) An earlier version of some of the ideas used in this paper and, in particular, the use of the O’Nan–Scott decomposition of primitive permutation groups, can be found in Sections 6 and 7 of Bosma and Cannon (1992). Recall that the socle of a finite group G is defined to be the subgroup generated by all of its minimal normal subgroups. It is isomorphic to a direct product of simple groups. Let 1 = G0 ≤ G1 ≤ G2 ≤ · · · ≤ Gr = G be a strictly ascending series of subgroups of G. If each Gi is normal in Gi+1 and Gi+1/Gi is simple, then it is called a composition series of G and the Gi+1/Gi are the composition factors of G. If each Gi is normal in G and Gi+1/Gi is a minimal normal subgroup of G/Gi, then it is called a chief series of G and the Gi+1/Gi are the chief factors of G. In this case, each chief factor is a direct product of isomorphic simple groups. Note that a chief series can be easily refined to