Results 1  10
of
292
The CRC Handbook Of Combinatorial Designs
, 1995
"... Introduction A group (P; \Delta) is a set P , together with a binary operation \Delta on P , for which 1. an identity element e 2 P exists, i.e. x \Delta e = e \Delta x = e for all x 2 P ; 2. \Delta is associative, i.e. x \Delta (y \Delta z) = (x \Delta y) \Delta z for all x; y; z 2 P ; 3. every el ..."
Abstract

Cited by 91 (2 self)
 Add to MetaCart
Introduction A group (P; \Delta) is a set P , together with a binary operation \Delta on P , for which 1. an identity element e 2 P exists, i.e. x \Delta e = e \Delta x = e for all x 2 P ; 2. \Delta is associative, i.e. x \Delta (y \Delta z) = (x \Delta y) \Delta z for all x; y; z 2 P ; 3. every element x 2 P has an inverse, an element x \Gamma1 for which x \Delta x \Gamma1 = x \Gamma1 \Delta<F25.
The inverse Galois problem and rational points on moduli spaces
 Math. Annalen
, 1991
"... Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteri ..."
Abstract

Cited by 57 (25 self)
 Add to MetaCart
Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p. Many attempts have been made to realize finite groups as Galois groups of extensions of Q(x) that are regular over Q (see the end of this introduction for definitions). We call this the “regular inverse Galois problem. ” We show that to each finite group G with trivial center and integer r ≥ 3 there is canonically associated an algebraic variety, Hin r (G), defined over Q (usually reducible) satisfying the following.
Random walk in a Weyl chamber
 Proc. Amer. Math. Soc
, 1992
"... Abstract. The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge xx> X2> ■> x„ (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps. To any simple and ..."
Abstract

Cited by 39 (4 self)
 Add to MetaCart
Abstract. The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge xx> X2> ■> x„ (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps. To any simple and natural proof, one can ask the question: How far can it be generalized? We will attempt to give one possible answer to this question for Andre's [A] celebrated solution of the twocandidate ballot problem. Andre's proof uses a reflection argument, and we will show that it can be naturally generalized
A Residue Calculus for Root Systems
, 2000
"... Let V be a finite dimensional real vector space on which a root system is given. Consider a meromorphic function ' on V C = V +iV , the singular locus of which is a locally finite union of hyperplanes of the form f 2 V C j h; i = sg, 2 , s 2 R. Assume ' is of suitable decay in the imaginary direct ..."
Abstract

Cited by 31 (13 self)
 Add to MetaCart
Let V be a finite dimensional real vector space on which a root system is given. Consider a meromorphic function ' on V C = V +iV , the singular locus of which is a locally finite union of hyperplanes of the form f 2 V C j h; i = sg, 2 , s 2 R. Assume ' is of suitable decay in the imaginary directions, so that integrals of the form '() d make sense for generic 2 V . A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and PaleyWiener theorems on semisimple symmetric spaces.
Refined anisotropic Ktypes and supercuspidal representations
 Pacific J. Math
, 1998
"... Let F be a nonarchimedean local field, and G a connected reductive group defined over F. We classify the representations of G(F) that contain any anisotropic unrefined minimal Ktype satisfying a certain tameness condition. We show that these representations are induced from compact (mod center) sub ..."
Abstract

Cited by 30 (5 self)
 Add to MetaCart
Let F be a nonarchimedean local field, and G a connected reductive group defined over F. We classify the representations of G(F) that contain any anisotropic unrefined minimal Ktype satisfying a certain tameness condition. We show that these representations are induced from compact (mod center) subgroups, and we construct corresponding refined minimal Ktypes. 0. Introduction. Let G be any connected reductive group defined over a nonarchimedean local field F of residual characteristic p. Under some tameness assumptions on G, we construct families of positivedepth supercuspidal representations of G = G(F). In particular, we classify (§2.7) the representations of G
On the Diameter of Finite Groups
 SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1990
"... The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter wh ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if S is allowed to exceed (log G) 1+c (c > 0), it plays a heavy role when jSj = O(1). In particular, every nonabelian nite simple group has a set of 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index. { Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((n ln n) 1=2 (1 + o(1))) upper bound. In the average case, the upper bound improves to exp((ln n) 2 (1 + o(1))). As a rst step toward extending this result to simple groups other than An , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for An . In the limited space of this article, we try to illuminate some of the basic underlying techniques.
Short Presentations for Finite Groups
 JOURNAL OF ALGEBRA
, 1997
"... We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1) . We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity ..."
Abstract

Cited by 26 (11 self)
 Add to MetaCart
We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1) . We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name of G, assuming in the case of Lie type simple groups over GF (p m ) that an irreducible polynomial f of degree m over GF (p) and a primitive root of GF (p m ) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groups PSU(3; q) = 2 A 2 (q), the Suzuki groups Sz(q) = 2 B 2 (q), and the Ree groups R(q) = 2 G 2 (q). In particular, all finite groups G without composition factors of these types have presentations of length O((log jGj) 3 ). For...
Sylow's Theorem in Polynomial Time
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1985
"... Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylo ..."
Abstract

Cited by 24 (8 self)
 Add to MetaCart
Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylow psubgroups PI, P, of G there is an element go G such that Pz = g‘PI g. We present polynomialtime algorithms that find (generators for) a Sylow psubgroup of G, and that find ge G such that P, = g‘P, g whenever (generators for) two Sylow psubgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups. 0 1985 Academic Press. Inc. PART I 1.