Some Remarks on the Probability of Generating an Almost Simple Group

Abstract

Abstract.We compare the probability of generating with a given number of random elements two almost simple groups with the same socle S. In particular we analyse the case S = PSL(2, p).2000 Mathematics Subject Classification. 20P05, 20D06, 20D60. Introduction.A finite nonabelian simple group S can be identified with a subgroup of its automorphism group Aut S. In [1] it is proved that for any pair of elements g 1 , g 2 in Aut S, there exist s 1 , s 2 in S such that g 1 s 1 , g 2 s 2 = g 1 , g 2 , S , i.e. the subgroup of Aut S generated by g 1 s 1 , g 2 s 2 contains S. Given g 1 , g 2 in Aut S we want to study the probability P g 1 ,g 2 (S) that a pair of elements s 1 , s 2 satisfies the conditionFirst we need to recall some definitions. For any finite group let φ G (t) denote the number of ordered t-tuples (g 1 , . . . , g t ) of elements of G that generate G. The number|G| t gives the probability that t randomly chosen elements of G generate G. Moreover if N is a normal subgroup of G, we define P G,N (t) = P G (t)/P G/N (t). This number is the probability that a t-tuple generates G, given that it generates G modulo N.is precisely the number of t-tuples (n 1 , . . . , n t ) ∈ N t such that G = g 1 n 1 , . . . , g t n t . In our particular case P g 1 ,g 2 (S) = P G,S (2), where G = g 1 , g 2 , S . Now define P(S) to be the probability that two randomly chosen elements of Aut S generate a subgroup containing S. We want to compare P(S) with P S (2). Note that