The probability of generating a finite simple group
نویسندگان
چکیده
منابع مشابه
On Probability of Generating a Finite Group
Let G be a finite group, and let φk(G) be the probability that k random group elements generate G. Denote by θ(G) the smallest k such that φk(G) > 1/e. In this paper we analyze quantity θ(G) for different classes of groups. We prove that θ(G) ≤ κ(G)+ 1 when G is nilpotent and κ(G) is the minimal number of generators of G. When G is solvable we show that θ(G) ≤ 3.25κ(G) + 107. We also show that ...
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In [KL] it is provcd that the probability of two randomly chosen elements of a finite dassical simple group G actually generating G tends to 1 as lei increases. If gEe, let Pu(g) be the probability that, if h is chosen randomly in G, then (g, h) IG. Let PG : = ma..x{Pu(g) I 9 E e#}. In [KL, Conjecture 2] it is suggested that a stronger result might hold: Pc ---4 0 as IGI ---4 00 for simple clas...
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In [2], Dixon considered the following question: " Suppose two permutations are chosen at random from the symmetric group Sn of degree n. What is the probability that they will generate Sn? " Actually, Netto conjectured last century that almost all pairs of elements from Sn will generate Sn or An. Dixon showed that this is true in the following sense: The proportion of ordered pairs (x, y) (x, ...
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We prove that the product replacement graph on generating k-tuples of a simple group contains a large connected component if k ≥ 3. This is related to the recent conjecture of Diaconis and Graham. As an application, we also prove that the output of the product replacement algorithm (see [CLMNO]) in this case does not have a strong bias.
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ژورنال
عنوان ژورنال: Israel Journal of Mathematics
سال: 2013
ISSN: 0021-2172,1565-8511
DOI: 10.1007/s11856-013-0034-7