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Blackbox recognition of finite simple groups of Lie type by statistics of element orders
 JOURNAL OF GROUP THEORY
, 2002
"... Given a blackbox group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses ..."
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Cited by 13 (5 self)
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Given a blackbox group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses a relatively small number...
Fast constructive recognition of black box orthogonal groups
 JOURNAL OF ALGEBRA
, 2006
"... We present an algorithm that constructively recognises when a given black box group is a nontrivial homomorphic image of the orthogonal group Ω ε (d, q) for known ε, d and q . The algorithm runs in polynomial time assuming oracles for handling SL(2, q) subgroups and discrete logarithms in F∗. ..."
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Cited by 7 (1 self)
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We present an algorithm that constructively recognises when a given black box group is a nontrivial homomorphic image of the orthogonal group Ω ε (d, q) for known ε, d and q . The algorithm runs in polynomial time assuming oracles for handling SL(2, q) subgroups and discrete logarithms in F∗.
Black box exceptional groups of Lie type
 In preparation
, 2002
"... If a black box group is known to be isomorphic to an exceptional simple group of Lie type of rank> 1, other than any 2 F4(q), over a field of known size, a Las Vegas algorithm is used to produce a constructive isomorphism. This yields an upgrade of all known nearly linear time Monte Carlo permutatio ..."
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Cited by 5 (2 self)
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If a black box group is known to be isomorphic to an exceptional simple group of Lie type of rank> 1, other than any 2 F4(q), over a field of known size, a Las Vegas algorithm is used to produce a constructive isomorphism. This yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to a rank 1 group or to any 2 F4(q). 1
Computing with Matrix Groups
 GROUPS, COMBINATORICS AND GEOMETRY
, 2001
"... A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting dis ..."
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Cited by 4 (4 self)
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A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting discussed here is as follows: a group is given, speciﬁed as G = X in terms of some generating set X of its elements, where X is an arbitrary subset of either Sn or GL(d, q ) (a familiar example is the group of Rubik’s cube). The goal is then to ﬁnd properties of G eﬃciently, such as G, the derived series, a composition series, Sylow subgroups, and so on.
INVARIABLE GENERATION AND THE CHEBOTAREV INVARIANT OF A FINITE GROUP
"... Abstract. A subset S of a finite group G invariably generates G if G = 〈s g(s)  s ∈ S 〉 for each choice of g(s) ∈ G, s ∈ S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in [KZ] we also bound the size of ..."
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Cited by 2 (2 self)
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Abstract. A subset S of a finite group G invariably generates G if G = 〈s g(s)  s ∈ S 〉 for each choice of g(s) ∈ G, s ∈ S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements. Dedicated to Bob Guralnick in honor of his 60th birthday 1.
On the number of pregular elements in finite simple groups
, 2007
"... A pregular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is pregular. In particular, we show that the proportion of pregular elements in a finite classical si ..."
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Cited by 1 (0 self)
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A pregular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is pregular. In particular, we show that the proportion of pregular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n − 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27 √ n), and for sporadic simple groups, at least 2/29. We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of pregular elements in S is at least min{1/31, 1/(2n)}. Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998). Our result shows that in finite simple groups, pregular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomialtime Monte Carlo algorithms for matrix groups. 1