On the probability of generating a minimal <i>d</i>-generated group

Volta, Francesca Dalla; Lucchini, Andrea; Morini, Fiorenza

doi:10.1017/s1446788700002822

Cited by 15 publications

(17 citation statements)

References 8 publications

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“…In this section suppose that p is a prime, p ≥ 5, S = PSL(2, p) and G = Aut S. Since S and G are the only subgroups of Aut S containing S, our question about P(S) reduces to comparing φ S (2) and φ G/S (2). This can be done using Corollary 6.…”

Section: Psl(2 P)mentioning

confidence: 99%

“…In the following tables we compare the values of ψ S (2) and ψ Aut S (2) in some examples (these numbers may be found using GAP [3] …”

Section: Other Simple Groupsmentioning

confidence: 99%

“…In the particular case when |Aut S : S| = 2 then g 1 , g 2 , S is either S or Aut S; since P Aut S/S (2) = 3/4, we obtain P(S) = P S,S (2) 4 + 3P Aut S,S (2) 4 .…”

mentioning

confidence: 90%

“…By [2] Corollary 8, when u ≥ 2 we have d(G t ) ≤ u if and only if t ≤ ψ G (u). Suppose that S X, Y ≤ Aut S; if we want to compare the two growth sequences {d(X t )} t∈‫ގ‬ and {d(Y t )} t∈‫ގ‬ we need to compare the two functions ψ X and ψ Y ; in the particular case when Aut S/S is cyclic of prime order we want to compare ψ S (u) = φ S (u)…”

mentioning

confidence: 94%

“…S P S,S (2) P Aut S,S (2) P(S) Alt (5) 19 These examples show that in some cases the two numbers P(S) and P S (2) coincide, in other cases they can be different. In this paper we try to explain these phenomena, and in particular we study the case when S = PSL(2, p) proving A related question is the following: if S G ≤ Aut S and S is a finite nonabelian simple group we define ψ G (u) as follows:…”

mentioning

confidence: 98%

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Some Remarks on the Probability of Generating an Almost Simple Group

Volta¹,

Lucchini²,

Morini³

2003

Glasgow Math. J.

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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

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Section: Psl(2 P)mentioning

confidence: 99%

“…In the following tables we compare the values of ψ S (2) and ψ Aut S (2) in some examples (these numbers may be found using GAP [3] …”

Section: Other Simple Groupsmentioning

confidence: 99%

“…In the particular case when |Aut S : S| = 2 then g 1 , g 2 , S is either S or Aut S; since P Aut S/S (2) = 3/4, we obtain P(S) = P S,S (2) 4 + 3P Aut S,S (2) 4 .…”

mentioning

confidence: 90%

mentioning

confidence: 94%

mentioning

confidence: 98%

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