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60
Finite permutation groups and finite simple groups
 Bull. London Math. Soc
, 1981
"... In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been ..."
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Cited by 92 (3 self)
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In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of
Computing the Composition Factors of a Permutation Group in Polynomial Time
 Combinatorica
, 1987
"... Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors. ..."
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Cited by 21 (2 self)
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Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.
A modular formalisation of finite group theory
 In TPHOLs
, 2007
"... Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a longterm effort to formalise FeitThompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most ..."
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Cited by 18 (6 self)
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Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a longterm effort to formalise FeitThompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most compositional way. 1
A brief history of the classification of finite simple groups
 BAMS
"... Abstract. We present some highlights of the 110year project to classify the finite simple groups. ..."
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Cited by 16 (0 self)
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Abstract. We present some highlights of the 110year project to classify the finite simple groups.
The Status of the Classification of the Finite Simple Groups
 Mathematical Monthly
, 2004
"... Common wisdom has it that the theorem classifying the finite simple groups was proved around 1980. However, the proof of the Classification is not an ordinary proof because of its length and complexity, and even in the eighties it was a bit controversial. Soon after the theorem was established, Gore ..."
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Cited by 16 (0 self)
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Common wisdom has it that the theorem classifying the finite simple groups was proved around 1980. However, the proof of the Classification is not an ordinary proof because of its length and complexity, and even in the eighties it was a bit controversial. Soon after the theorem was established, Gorenstein, Lyons, and Solomon (GLS) launched a program to simplify large parts of the proof and, perhaps of more importance, to write it down clearly and carefully in one place, appealing only to a few elementary texts on finite and algebraic groups and supplying proofs of any “wellknown” results used in the original proof, since such proofs were scattered throughout the literature or, worse, did not even appear in the literature. However, the GLS program is not yet complete, and over the last twenty years gaps have been discovered in the original proof of the Classification. Most of these gaps were quickly eliminated, but one presented serious difficulties. The serious gap has recently been closed, so it is perhaps a good time to review the status of the Classification. I will begin slowly with an introduction to the problem and with some motivation. Recall that a group G is simple if 1 and G are the only normal subgroups of G; equivalently G ∼ =G/1 and 1 ∼ =G/G are the only factor groups
kHomogeneous Groups
 MATH Z.
, 1972
"... A permutation group is called khomogeneous if it is transitive on the ksets of permuted points. Theorem 1. Let G be a group khomogeneous but not ktransitive on a finite set f2 of n points, where n>=2k. Then, up to permutation isomorphism, one of the ..."
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Cited by 10 (3 self)
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A permutation group is called khomogeneous if it is transitive on the ksets of permuted points. Theorem 1. Let G be a group khomogeneous but not ktransitive on a finite set f2 of n points, where n>=2k. Then, up to permutation isomorphism, one of the
The history of qcalculus and a new method
, 2000
"... 1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8 ..."
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Cited by 10 (8 self)
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1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8
On 2Transitive groups in which the stabilizer of two points Fixes Additional Points
 JOURNAL OF THE LONDON MATHEMATICAL SOCIETY
, 1972
"... Let F be a 2transitive group of finite degree v such that the stabilizer Txy of the distinct points x and y fixes precisely k points, where 2 < k < v. The only known nonsolvable groups with this property are of the following types: (i) a 2transitive collineation group of PG(d, 2) for some d; (ii) ..."
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Cited by 10 (4 self)
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Let F be a 2transitive group of finite degree v such that the stabilizer Txy of the distinct points x and y fixes precisely k points, where 2 < k < v. The only known nonsolvable groups with this property are of the following types: (i) a 2transitive collineation group of PG(d, 2) for some d; (ii) a 2transitive collineation group of
Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Cited by 7 (7 self)
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
Finite Bruck loops
"... Let X be a magma; that is X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y ↦ → y ◦ x and L(x) : y ↦ → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are ..."
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Cited by 7 (3 self)
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Let X be a magma; that is X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y ↦ → y ◦ x and L(x) : y ↦ → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are permutations of X for all x ∈ X. In essence loops are groups without the associative axiom. See [Br, Pf] for further discussion of basic properties of loops. Certain classes of loops have received special attention: A loop X is a (right) Bol loop if it satisfies the (right) Bol identity (Bol): (Bol) or equivalently (Bol2) ((z ◦ x) ◦ y) ◦ x = z ◦ ((x ◦ y) ◦ x). R(x)R(y)R(x) = R((x ◦ y) ◦ x). for all x, y, z ∈ X. In a Bol loop, the subloop 〈x 〉 generated by x ∈ X is a group. Thus we can define x −1 and the order x  of x to be, respectively, the inverse of x and the