## An Atlas of characteristic zero representations (2005)

Citations: | 1 - 0 self |

### BibTeX

@TECHREPORT{Nickerson05anatlas,

author = {Simon Jon Nickerson},

title = {An Atlas of characteristic zero representations},

institution = {},

year = {2005}

}

### OpenURL

### Abstract

Motivated by the World Wide Web Atlas of Finite Group Representations and the re-cent classification of low-dimensional representations of quasisimple groups in cross-characteristic fields by Hiss and Malle, we construct with a computer over 650 rep-resentations of finite simple groups. Explicit matrices for these representations are available on the Internet and are included on an attached CD-ROM. Our main tool is a GAP program for decomposing permutation modules. It uses reduction modulo various primes and rational reconstruction to give an acceptable performance. In addition, we define standard generators for the groups under consideration, and exhibit black box algorithms for finding standard generators and checking whether given elements of the group are standard generators. To my parents Acknowledgements I have benefited greatly from the guidance and support of my supervisor, Professor Robert Wilson. I wish to thank him for his encouragement and enthusiasm in this project. I feel privileged to have been one of his students. I am indebted to my examiners Professor Derek Holt and Dr Paul Flavell for their detailed reading of the text and for pointing out several improvements. I thank Dr John Bray who has very helpfully shared his knowledge of computational group theory with me. He has also provided two interesting representations for inclusion here. I also thank Richard Barraclough for helping me with the Monster group programs and Dr Frank Lübeck for helping me with my questions about CHEVIE. My work has been made greatly easier by the GAP computer algebra system. I thank all the developers for their hard work in producing such a marvellous tool and making it freely available. I am very grateful to Sophie Whyte, Elizabeth Wharton and Marijke van Gans, with whom I have shared an office for the past three years. I wish to thank them for many useful conversations, for helping with the crossword, and for patiently putting up with my annoying habits. I also thank the School of Mathematics at the University of Birmingham for providing a stimulating environment for research.

### Citations

1240 | The C Programming Language
- Kernighan, Ritchie
(Show Context)
Citation Context ... storage to perform calculations with such enormous matrices. Instead, we use the computer construction by Linton et al. [31] with an implementation by Parker and Wilson in the C programming language =-=[28]-=-. We will give an overview of the way we use this construction. The Monster group M acts linearly on a 196882-dimensional vector space V over GF(2). The group is generated by a subgroup: H = 〈A, B, C,... |

385 |
Atlas of Finite Groups
- Conway
- 1985
(Show Context)
Citation Context ...tions available . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.1 Predicates for the BBOX language . . . . . . . . . . . . . . . . . . . . . . 207sNotation and Conventions We follow ATLAS =-=[11]-=- conventions for naming groups, conjugacy classes, characters and representations. Composition series for groups are written ‘from the bottom up’. Names of computer programs and systems are set in san... |

349 |
The On-line Encyclopedia of Integer Sequences, published electronically at http://oeis.org
- Sloane
- 2013
(Show Context)
Citation Context ...e are many solutions to this pair of equations. The smallest values of n for which equations (3.4.16) and (3.4.17) have a solution are given in Table 3.3, and can also be found as sequence A108157 in =-=[47]-=-. The smallest value is n = 497. (A4) The 3 points are contained in three cycles (1 point in each). lcm(r, s, t) = n − 2 (3.4.20) r + s + t = n (3.4.21) Suppose r, s and t are all proper divisors of n... |

213 |
Duality for representations of a reductive group over a finite field
- Deligne, Lusztig
- 1982
(Show Context)
Citation Context ...ial amount of effort if we treat the groups individually. We can use whichever method is easiest to find each representation, without having to consider any other representations. Deligne and Lusztig =-=[14]-=- provide a uniform treatment for 2sthe groups of Lie type for those who require it. 0.2 The Hiss-Malle classification Gerhard Hiss and Gunter Malle have classified absolutely irreducible representatio... |

200 | Go to statement considered harmful
- Dijkstra
- 1968
(Show Context)
Citation Context ...ached, when the program moves to the appropriate ‘lbl’ command. At a ‘return’, the program returns to the location of the most recent ‘call’. Unconditional jumps are unfashionable in computer science =-=[15]-=-, but this type of flow control was preferred to more structured commands (such as do . . .while or repeat . . .until) because it is frequently important to be able to go back to previous steps if we ... |

187 |
Character theory of finite groups
- Isaacs
- 1994
(Show Context)
Citation Context ...To emphasise the fact that we are doing more than making a definition, we will label each definition by ‘Definition-Theorem’. 1.4 Structure constants In this section, we introduce structure constants =-=[25, 26]-=-. These will be useful when looking for characterisations of standard generators later in this chapter. Let G be a finite group with complex group algebra CG. Let the conjugacy classes of G be C (1) ,... |

127 |
Theory of Groups of Finite Order
- Burnside
- 1911
(Show Context)
Citation Context ...ee) permutation characters of G. • Find a subgroup H of G whose permutation representation gives rise to that permutation character. (In particular, we know the order of H.) Definition 8.16 (Burnside =-=[8]-=-) A table of marks for a group G is a matrix M whose rows and columns are labelled by conjugacy classes of subgroups of G, and given subgroups H, K of G in conjugacy classes H G , K G respectively, th... |

98 |
The subgroup structure of the finite classical groups
- Kleidman, Liebeck
- 1990
(Show Context)
Citation Context ...g shapes: StabG(isotropic) ∼ = q 3 .Ω3(q) ∼ = q 3 .L2(q) (5.2.1) StabG(plus type) ∼ = q 3 .Ω + 4 (q) ∼ = 2·(L2(q) × L2(q)) (5.2.2) StabG(minus type) ∼ = q 3 .Ω − 4 (q) ∼ = L2(q 2 ) (5.2.3) Proof. See =-=[29]-=- � 97sProof (Lemma 5.8). As above, we will think of S4(q) = PSp 4 (q) as Ω5(q), so z1 and z2 are endomorphisms of a 5-dimensional space V over GF(q) equipped with a nondegenerate quadratic form q : V ... |

97 |
zur Gathen and Jürgen Gerhard. Modern Computer Algebra
- von
- 1999
(Show Context)
Citation Context ... The rest of this section will be devoted to step 2 of the above procedure. Recall that we are trying to solve the congruence y ≡ m (mod N) where N = ∏p∈P p. We apply the Extended Euclidean Algorithm =-=[54]-=- to N and m as follows. Set n0 = N, n1 = m, and perform the Euclid’s algorithm for calculating gcd(N, m): n0 = q1n1 + n2 n1 = q2n2 + n3 . . nr−1 = qrnr + nr+1 nr = qr+1nr+1 + 0 For each 2 ≤ i ≤ r + 1 ... |

92 |
Schnelle Multiplikation grosser Zahlen
- Schönhage, Strassen
- 1971
(Show Context)
Citation Context ... gcd. Adding integers cannot be done faster than O(k), but there exist algorithms for gcd which are O(k 2 / log k) [12, 49] and an algorithm for multiplying two integers which is O(k log k log log k) =-=[44]-=-. 153sare represented by an integer 0 ≤ i < q − 1 called the logarithm of x. The logarithm i satisfies x = ω i (where ω is a fixed primitive element of GF(q)). The zero element is represented by ω −∞ ... |

70 | CHEVIE – A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras
- Geck, Hiss, et al.
- 1996
(Show Context)
Citation Context ... the GAP functionality was too slow (in particular when dealing with finite fields) and we had to find some alternative. Other notable computer systems that we used during this project were: • CHEVIE =-=[20]-=-. This is a package for Maple [53] for performing character table calculations for ‘generic’ groups of Lie type (i.e. calculations which depend only on the Dynkin diagram for the group, not the order ... |

58 |
An Atlas of Brauer Characters
- Jansen, Parker, et al.
- 1995
(Show Context)
Citation Context ...red: • Matrices purporting to generate a 483-dimensional representation of M23 over GF(7) were included, but they failed to satisfy the semi-presentation. In fact no such representation of M23 exists =-=[27]-=-. • One of the 896-dimensional representations of HS over GF(4) was incorrect, as 51sthe product of the two generators had order exceeding 100. • Matrices purporting to generate a 104-dimensional repr... |

57 |
Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschften, Band 134
- Huppert
- 1967
(Show Context)
Citation Context ...s for the groups in L G of the form L2(q). 4.1 Basic facts about L2(q) Let G = PSL2(q) = L2(q) for q = p r a prime power. We will require the following facts about G in the sequel: Lemma 4.1 (Dickson =-=[24]-=-) Any maximal subgroup of G must be one of the following: • a Borel subgroup: the semi-direct product of C r p by a cyclic group of order (q − 1)/2 (or q − 1 if q is even) • a dihedral group • a subfi... |

51 |
The computer calculation of modular characters (the meataxe).’ ‘Computational group theory
- Parker
- 1982
(Show Context)
Citation Context ...sentatives. The Web Atlas (as it is often called) contains information on all the finite simple groups featured in the ATLAS [11] as well as some larger groups. 1sBecause of such tools as the MeatAxe =-=[36, 41]-=- of Parker, the Web Atlas has fairly good coverage of modular representations. However, the Web Atlas has relatively few representations over fields of characteristic zero. The main reason for this is... |

40 |
Untersuchung uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen
- Schur
- 1907
(Show Context)
Citation Context .../(p + 1) if p ≡ −1 (mod 12) ⎪⎩ 1 if p ≡ ±5 (mod 12) (4.2.8) In order to calculate β, we need a character table for L2(p). This can be read from the character table of SL2(p), which was known by Schur =-=[45]-=-, and is given in [17, section 38]. The relevant character entries are given in Table 4.3, which uses the following notational device: Notation 4.9 We adopt the following notation for periodic functio... |

31 |
The symmetric group, The Wadsworth & Brooks/Cole
- Sagan
- 1991
(Show Context)
Citation Context ...resentations of Sn. In this section, we will provide a summary of the facts we need, as well as a construction of the irreducible representations of Sn. An exposition of this material can be found in =-=[43]-=-. 132sDefinition 7.1 A partition of n is a non-increasing tuple λ = (λ1, λ2, . . . λr) of positive integers which sums to n. We write λ ⊣ n. Such a tuple defines subsets Ω i ⊆ Ω for 1 ≤ i ≤ r as follo... |

24 |
Two Fast GCD Algorithms
- Sorenson
- 1994
(Show Context)
Citation Context ... arbitrary integers. Multiplying rational numbers requires multiplying and taking the gcd. Adding integers cannot be done faster than O(k), but there exist algorithms for gcd which are O(k 2 / log k) =-=[12, 49]-=- and an algorithm for multiplying two integers which is O(k log k log log k) [44]. 153sare represented by an integer 0 ≤ i < q − 1 called the logarithm of x. The logarithm i satisfies x = ω i (where ω... |

23 | Group Representation Theory, Part A - Dornhoff - 1971 |

20 | Fast constructive recognition of a black box group isomorphic to Sn or An using Goldbach conjecture
- Bratus, Pak
(Show Context)
Citation Context ... We will then exhibit finders for these groups for the case n ≤ 25. Black box algorithms for constructive recognition of symmetric and alternating groups have also been investigated by Bratus and Pak =-=[5]-=- and Beals et al [2]. 3.1 Standard generators for Sn We begin with the symmetric groups, where the analysis is somewhat simpler. Let G = Sn, and let: x = (1, 2), y = (2, 3, 4 . . . n) (3.1.1) We note ... |

19 | Standard generators for sporadic simple groups - Wilson - 1996 |

18 | Calculating the order of an invertible matrix
- Celler, Leedham-Green
- 1997
(Show Context)
Citation Context ...ned. We are interested in the problem of finding specific standard generators in a group isomorphic to G. 4 The oracle for calculating orders of elements g ∈ GLd(q) is due to Celler and Leedham-Green =-=[10]-=-. The order of g can be calculated in O(d 3 log q) time providing the numbers q i − 1 for 1 ≤ i ≤ d can be factorised. 21sDefinition 1.7 A finder for G is a black box algorithm which, given a black bo... |

14 |
Low-dimensional representations of quasi-simple groups
- Hiß, Malle
(Show Context)
Citation Context ...s ρ : G → GL d(k) where G is a finite quasisimple group, the dimension d is at most 250 and the characteristic of the field k differs from the defining characteristic of G if G is a group of Lie type =-=[21, 22]-=-. From their work, we can easily extract a classification of the irreducible complex representations of the finite simple groups whose dimension does not exceed 250. Our discussion in this section wil... |

13 |
The Monster is a Hurwitz group
- Wilson
(Show Context)
Citation Context ...it must be a 3B-element. To find standard generators, we powered up a representative of class 4B from [1] to give an involution x, and then looked for conjugates y of a 3B class representative b from =-=[59]-=- such that o(xy) = 29. (We will be able to prove retrospectively that x and y 44sare in the correct classes.) We used: x = (DC 3 D 2 CD 2 CD 2 CD 2 CDCDCDC) 2 b = (ABABAB 2 AB) 7 y = ((ABABAB 2 AB) 7 ... |

12 |
The character of the finite symplectic group Sp(4, q
- Srinivasan
- 1968
(Show Context)
Citation Context ...5.1.5) ξG(2B, 3B, C ) = 2 (5.1.6) ξG.2(2B, 3A, C ) = 1 (5.1.7) ξG.2(2B, 3B, C ) = 1 (5.1.8) Proof. 1. Examining the summation formula for structure constants (1.4.7) and the character table for S4(q) =-=[50]-=- 1 we see that most of the terms in the sum are zero. For those characters where there is a non-zero contribution, the values of the character on 3A and 3B agree, so it suffices to calculate the struc... |

11 | Acceleration of Euclidean Algorithm and Rational Number Reconstruction
- Wang, Pan
- 2007
(Show Context)
Citation Context ...e with the smallest height is xP = 3/14. Remark Using a more complicated algorithm, rational reconstruction can be achieved in O(k log 2 k log log k) time, where k is the number of binary digits in N =-=[56]-=-. The algorithm described above is O(k 2 ). 160s8.13 Reconstruction of quadratic elements The principles involved in rational reconstruction can apply to infinite fields other than Q. We extended the ... |

8 |
Ákos Seress. A black-box group algorithm for recognizing finite symmetric and alternating groups
- Beals, Leedham-Green, et al.
(Show Context)
Citation Context ...t finders for these groups for the case n ≤ 25. Black box algorithms for constructive recognition of symmetric and alternating groups have also been investigated by Bratus and Pak [5] and Beals et al =-=[2]-=-. 3.1 Standard generators for Sn We begin with the symmetric groups, where the analysis is somewhat simpler. Let G = Sn, and let: x = (1, 2), y = (2, 3, 4 . . . n) (3.1.1) We note that x is a transpos... |

8 |
Maximal overgroups of Singer elements in classical groups
- Bereczky
(Show Context)
Citation Context ...d xy has order (q 2 + 1)/2. In order to show that each semi-standard pair is a generating pair for S4(q), we will need the following definition and lemma. Definition 5.2 We define (following Bereczky =-=[3]-=-): • A Singer subgroup of GLn(q) is a cyclic subgroup of order q n − 1. • Let G ≤ GLn(q) be a finite classical group. A Singer subgroup of G is an irreducible cyclic subgroup of G of maximal possible ... |

8 |
Constructing representations of finite groups
- Dixon
- 1993
(Show Context)
Citation Context ...a method for producing representations affording a character χ of a group G subject to the condition that there exists a subgroup H ≤ G such that χH has a constituent with degree 1 and multiplicity 1 =-=[16]-=-. Unfortunately, finding such a subgroup H is not easy in general (it seems to require computing the whole lattice of subgroups), and there exist groups where no such subgroup exists. Dixon’s method h... |

8 |
Construction and Classification of Irreducible Representations of Special Linear Group of the Second Order over a Finite Field
- Tanaka
- 1967
(Show Context)
Citation Context ...− 1)-dimensional representations of L2(q) is more complicated than the others. We will sketch the construction here, following Piatetski-Shapiro [39], although the construction is described elsewhere =-=[38, 52]-=-. Let K = GF(q) and let L = GF(q 2 ) be the (unique) quadratic extension of K. We fix a non-trivial character Ψ of the additive group K (an elementary abelian p-group). Let V be the (q − 1)-dimensiona... |

6 | An algorithm for constructing representations of finite groups
- Dabbaghian-Abdoly
- 2005
(Show Context)
Citation Context ... is not easy in general (it seems to require computing the whole lattice of subgroups), and there exist groups where no such subgroup exists. Dixon’s method has been investigated by Dabbaghian-Abdoly =-=[13]-=- and implemented by him as a GAP package Repsn. The method is quite slow and memory-intensive, but it succeeded where other methods failed (notably for some small representations of the unitary groups... |

6 |
Representations and characters of groups. Cambridge Mathematical Textbooks
- James, Liebeck
- 1993
(Show Context)
Citation Context ...To emphasise the fact that we are doing more than making a definition, we will label each definition by ‘Definition-Theorem’. 1.4 Structure constants In this section, we introduce structure constants =-=[25, 26]-=-. These will be useful when looking for characterisations of standard generators later in this chapter. Let G be a finite group with complex group algebra CG. Let the conjugacy classes of G be C (1) ,... |

6 | Constructing rational representations of finite groups
- Plesken, Souvignier
- 1996
(Show Context)
Citation Context ... can use representations of ¯G in the construction as well. This is frequently useful if ¯G has representations of smaller dimension than G. 10.2.1 Plesken-Souvignier splitting Plesken and Souvignier =-=[40]-=- describe a way of decomposing a rational representation into homogeneous parts. We have written a program in C [28] which uses their technique. Let V be a QG-module which decomposes into irreducibles... |

5 |
Finding possible permutation characters
- Breuer, Pfeiffer
- 1998
(Show Context)
Citation Context ... the first part has been effectively solved already, as it is easy to extract a complete list of permutation characters from a table of marks. Otherwise, we can use the methods of Breuer and Pfeiffer =-=[7]-=- to find ‘possible’ permutation characters. These are characters of G which obey a set of necessary conditions for a character to be a permutation character, and their methods have been implemented as... |

5 | Permutation group algorithms. Number 152 in Cambridge tracts in mathematics - Seress - 2002 |

4 | Abstract definitions of the symmetric and alternating groups and certain other permutation groups. The quarterly journal of pure and applied mathematics 49 - Carmichael - 1923 |

4 |
The C MeatAxe, Release 1.5. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule
- Ringe
- 1993
(Show Context)
Citation Context ...sentatives. The Web Atlas (as it is often called) contains information on all the finite simple groups featured in the ATLAS [11] as well as some larger groups. 1sBecause of such tools as the MeatAxe =-=[36, 41]-=- of Parker, the Web Atlas has fairly good coverage of modular representations. However, the Web Atlas has relatively few representations over fields of characteristic zero. The main reason for this is... |

3 |
F and other simple groups
- Norton
- 1975
(Show Context)
Citation Context ...nging a basis). 140sGroup Dimension Indicator Source U5(2) 10 − ATLAS G2(3) 14 + ATLAS 2 F4(2) ′ 26 ◦ ATLAS (restricted from 2.F4(2)) M24 45 ◦ [32, 42] Fi22 78 + ATLAS Sz(32) 124 ◦ John Bray HN 133 + =-=[6, 35]-=- Th 248 + [48] Table 7.3: Representations from other sources 141sChapter 8 Split-P: a GAP system for splitting permutation modules In this chapter, we describe our Split-P system, which is a set of GA... |

3 | An integral meataxe, in: The atlas of finite groups: ten years on - Parker - 1995 |

3 |
representations of the symplectic group
- Szechtman, Weil
- 1998
(Show Context)
Citation Context ...entations of symplectic groups Let G = Sp 2n (q) with q = p r odd. We can construct a q n -dimensional representation called the Weil representation as follows. (We follow the discussion in Szechtman =-=[51]-=-.) Let K = GF(q) and let V be a 2n-dimensional K-vector space equipped with a nondegenerate symplectic form 〈 〉. Then we can define a group H to be the set: with multiplication given by: Then H is a g... |

2 |
A Stronger Bertrand’s Postulate with an Application to Partitions
- Dressler
- 1972
(Show Context)
Citation Context ...= 6), the taming set T(Tn) is always non-empty. First of all, we need a lemma. Lemma 3.5 Any integer greater than 9 can be expressed as a sum of distinct odd primes. Proof. This is proved by Dressler =-=[18]-=- using induction and the following result: pn+1 < 2pn − 10 (n > 6) (3.2.2) which is a strengthening of Bertrand’s Postulate. � 56sTheorem 3.6 Let n ≥ 5, n �= 6 be an integer. Then: 1. There exists an ... |

2 |
The Meataxe as a tool in computational group theory, The atlas of finite groups: ten years on
- Holt
- 1995
(Show Context)
Citation Context ...has relatively few representations over fields of characteristic zero. The main reason for this is computational difficulty. Though Parker’s methods can be adapted to work with Z, Q or its extensions =-=[37, 23]-=-, there are theoretical and computational problems which make the endeavour a lot more challenging. In a finite field, splitting up a complicated module can be done providing one is willing to wait lo... |

2 |
Maximal subgroups of the sporadic almost-simple groups. MPhil (Qual
- Nickerson
- 2004
(Show Context)
Citation Context ... consistency with our calculations in G later.) Words for standard generators of the subgroup LK ∼ = L in terms of standard generators e, f of K were found by the methods described in my MPhil thesis =-=[34]-=-: g2 = e (9.4.3) h2 = ((e f e f e f e f f ) 3 e f e f f ) 9.5 Standard basis for standard generators of L (9.4.4) Let g, h denote standard generators of the 231-dimensional representation of L under c... |

2 |
Complex representations of GL(2, q
- Pergler
- 1995
(Show Context)
Citation Context ...− 1)-dimensional representations of L2(q) is more complicated than the others. We will sketch the construction here, following Piatetski-Shapiro [39], although the construction is described elsewhere =-=[38, 52]-=-. Let K = GF(q) and let L = GF(q 2 ) be the (unique) quadratic extension of K. We fix a non-trivial character Ψ of the additive group K (an elementary abelian p-group). Let V be the (q − 1)-dimensiona... |

2 |
Complex representations of GL(2, K) for finite fields K, volume 16 of Contemporary Mathematics
- Piatetski-Shapiro
- 1983
(Show Context)
Citation Context ...al representations of L2(q) The construction of the (q − 1)-dimensional representations of L2(q) is more complicated than the others. We will sketch the construction here, following Piatetski-Shapiro =-=[39]-=-, although the construction is described elsewhere [38, 52]. Let K = GF(q) and let L = GF(q 2 ) be the (unique) quadratic extension of K. We fix a non-trivial character Ψ of the additive group K (an e... |

1 |
Conjugacy class representatives in the Monster
- Barraclough, Wilson
- 2005
(Show Context)
Citation Context ... 31, 34, 50, 55, 68, 94}. The structure constants for M then imply that y is not a 3C-element, so it must be a 3B-element. To find standard generators, we powered up a representative of class 4B from =-=[1]-=- to give an involution x, and then looked for conjugates y of a 3B class representative b from [59] such that o(xy) = 29. (We will be able to prove retrospectively that x and y 44sare in the correct c... |

1 |
The MAGMA Handbook
- Bosma, Cannon
(Show Context)
Citation Context ...ystem was originally designed for computational group theory, although its more recent versions have support for other mathematical objects (such as rings and Lie algebras). The Magma computer system =-=[4]-=- is based in Sydney, Australia. Its development is headed by John Cannon. In some ways, Magma is a much more ambitious system than GAP, and it is much less tied to group theory calculations. 5sMost of... |

1 |
Monomial modular representations and symmetric generation of the Harada-Norton group
- Bray, Curtis
(Show Context)
Citation Context ...nging a basis). 140sGroup Dimension Indicator Source U5(2) 10 − ATLAS G2(3) 14 + ATLAS 2 F4(2) ′ 26 ◦ ATLAS (restricted from 2.F4(2)) M24 45 ◦ [32, 42] Fi22 78 + ATLAS Sz(32) 124 ◦ John Bray HN 133 + =-=[6, 35]-=- Th 248 + [48] Table 7.3: Representations from other sources 141sChapter 8 Split-P: a GAP system for splitting permutation modules In this chapter, we describe our Split-P system, which is a set of GA... |

1 |
A shift and cut GCD algorithm
- Cruz, Salzberg
- 1994
(Show Context)
Citation Context ... arbitrary integers. Multiplying rational numbers requires multiplying and taking the gcd. Adding integers cannot be done faster than O(k), but there exist algorithms for gcd which are O(k 2 / log k) =-=[12, 49]-=- and an algorithm for multiplying two integers which is O(k log k log log k) [44]. 153sare represented by an integer 0 ≤ i < q − 1 called the logarithm of x. The logarithm i satisfies x = ω i (where ω... |

1 |
Sums of distinct primes
- Kløve
- 1973
(Show Context)
Citation Context ...Rn) is non-empty. Proof. For n ≤ 30, consult Table 3.5. For n > 30, we can write: n − 3 = k ∑ pi i=1 (3.5.3) where the p i are distinct primes greater than or equal to 5 (this is by a result of Kløve =-=[30]-=- analogous to Lemma 3.5). Then set rn = 3 ∏ p i as before. Any element of Sn with this order must have k + 1 cycles of orders 3, p1, . . . p k respectively, and hence must be in An and must power up t... |

1 |
A geometry for M24
- Margolin
- 1993
(Show Context)
Citation Context ...ed in Table 7.3, we use these constructions (possibly changing a basis). 140sGroup Dimension Indicator Source U5(2) 10 − ATLAS G2(3) 14 + ATLAS 2 F4(2) ′ 26 ◦ ATLAS (restricted from 2.F4(2)) M24 45 ◦ =-=[32, 42]-=- Fi22 78 + ATLAS Sz(32) 124 ◦ John Bray HN 133 + [6, 35] Th 248 + [48] Table 7.3: Representations from other sources 141sChapter 8 Split-P: a GAP system for splitting permutation modules In this chapt... |