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15
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 55 (4 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Computing in Solvable Matrix Groups
, 1992
"... We announce methods for efficient management of solvable matrix groups over finite fields. We show that solvability and nilpotencecan betestedin polynomialtime. Such efficiency seems unlikely for membershiptesting, which subsumes the discretelog problem. However, assuming that the primes in jGj ..."
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Cited by 30 (1 self)
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We announce methods for efficient management of solvable matrix groups over finite fields. We show that solvability and nilpotencecan betestedin polynomialtime. Such efficiency seems unlikely for membershiptesting, which subsumes the discretelog problem. However, assuming that the primes in jGj (other than the field characteristic) arepolynomiallybounded, membershiptesting and many other computational problems areinpolynomial time. These problems include finding stabilizers of vectors and of subspaces and finding centralizers and intersections of subgroups. An application to solvable permutation groups puts the problem of finding normalizers of subgroups into polynomial time. Some of the results carry over directly to finite matrix groups over algebraic number fields# thus, testing solvability is in polynomial time, as is testing membership and finding Sylow subgroups.
The minimal base size of primitive solvable permutation groups
 University of Leicester Leicester
, 1996
"... A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. Answering a question of Pyber, we prove that all primitive solvable permutation groups have a base of size at most four. 1. ..."
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Cited by 18 (1 self)
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A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. Answering a question of Pyber, we prove that all primitive solvable permutation groups have a base of size at most four. 1.
PolynomialTime Versions of Sylow's Theorem
 JOURNAL OF ALGORITHMS
, 1988
"... Let G be a subgroup of S,, given in terms of a generating set of permutations, and let p be a prime divisor of 1 G 1. If G is solvableand, more generally, if the nonabelian composition factors of G are suitably restrictedit is shown that the following can be found in polynomial time: a Sylow psub ..."
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Cited by 9 (3 self)
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Let G be a subgroup of S,, given in terms of a generating set of permutations, and let p be a prime divisor of 1 G 1. If G is solvableand, more generally, if the nonabelian composition factors of G are suitably restrictedit is shown that the following can be found in polynomial time: a Sylow psubgroup of G containing a given psubgroup, and an element of G conjugating a given Sylow psubgroup to another. Similar results are proved for Hall subgroups of solvable groups and a version of the SchurZassenhaus theorem is obtained.
On Probability Of Generating A Finite Group
, 1999
"... Let G be a finite group, and let ' k (G) be the probability that k random group elements generate G. Denote by #(G) the smallest k such that ' k (G) ? 1=e. In this paper we analyze quantity #(G) for different classes of groups. We prove that #(G) (G) + 1 when G is nilpotent and (G) ..."
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Cited by 8 (5 self)
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Let G be a finite group, and let ' k (G) be the probability that k random group elements generate G. Denote by #(G) the smallest k such that ' k (G) ? 1=e. In this paper we analyze quantity #(G) for different classes of groups. We prove that #(G) (G) + 1 when G is nilpotent and (G) is the minimal number of generators of G. When G is solvable we show that #(G) 3:25 (G) + 10 7 . We also show that #(G) ! C log log jGj, where G is a direct product of simple nonabelian groups, and C is a universal constant. The work is motivated by the applications to the "product replacement algorithm" (see [CLMNO,P4]). This algorithm is an important recent innovation, designed to efficiently generate (nearly) uniform random group elements. Recent work by Babai and the author [BaP] showed that the output of the algorithm must have a strong bias in certain cases. The precise probabilistic estimates we obtain here, combined with a note [P3], give positive result, proving that no bias exists for...
CIRCULANT GRAPHS: RECOGNIZING AND ISOMORPHISM TESTING IN POLYNOMIAL TIME
"... Abstract. An algorithm is constructed for recognizing the circulant graphs and finding a canonical labeling for them in polynomial time. This algorithm also yields a cycle base of an arbitrary solvable permutation group. The consistency of the algorithm is based on a new result on the structure of S ..."
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Cited by 8 (0 self)
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Abstract. An algorithm is constructed for recognizing the circulant graphs and finding a canonical labeling for them in polynomial time. This algorithm also yields a cycle base of an arbitrary solvable permutation group. The consistency of the algorithm is based on a new result on the structure of Schur rings over a finite cyclic group. A finite graph 1 is said to be circulant if its automorphism group contains a full cycle, 2 i.e., a permutation the cycle decomposition of which consists of a unique cycle of full length. This means that the graph admits a regular cyclic automorphism group, and, consequently, is isomorphic to a Cayley graph over a cyclic group. In particular, any
Large Galois groups with applications to Zariski density. arXiv preprint arXiv:1312.3009
, 2013
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The minimal base size of a psolvable linear group, submitted
"... Abstract. Let V be a finite vector space over a finite field of order q and of characteristic p. Let G ≤ GL(V) be a psolvable completely reducible linear group. Then there exists a base for G on V of size at most 2 unless q ≤ 4 in which case there exists a base of size at most 3. The first statemen ..."
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Abstract. Let V be a finite vector space over a finite field of order q and of characteristic p. Let G ≤ GL(V) be a psolvable completely reducible linear group. Then there exists a base for G on V of size at most 2 unless q ≤ 4 in which case there exists a base of size at most 3. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Pálfy and Wolf is also given. Dedicated to the memory of Ákos Seress. 1.
Transitive projective planes and insoluble groups
, 2009
"... Suppose that a group G acts transitively on the points of P, a finite nonDesarguesian projective plane. We prove first that the Sylow 2subgroups of G are cyclic or generalized quaternion; we then prove that if G is insoluble then G/O(G) is isomorphic to SL2(5) or SL2(5).2. MSC(2000): 20B25, 51A35. ..."
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Suppose that a group G acts transitively on the points of P, a finite nonDesarguesian projective plane. We prove first that the Sylow 2subgroups of G are cyclic or generalized quaternion; we then prove that if G is insoluble then G/O(G) is isomorphic to SL2(5) or SL2(5).2. MSC(2000): 20B25, 51A35. 1
ON A CONJECTURE OF GLUCK
"... Dedicated to Professor Nguyễn H.V. Hưng on the occasion of his sixtieth birthday. Abstract. Let F(G) and b(G) respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group G. A wellknown conjecture of D. Gluck claims that if G is solvable then ..."
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Dedicated to Professor Nguyễn H.V. Hưng on the occasion of his sixtieth birthday. Abstract. Let F(G) and b(G) respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group G. A wellknown conjecture of D. Gluck claims that if G is solvable then G: F(G)  ≤ b(G)2. We confirm this conjecture in the case where F(G)  is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure. 1.