Improvements to the ATLAS – II

This list contains improvements not in Appendix 2 of “An Atlas of Brauer characters” (by Jansen, Lux, Parker & Wilson, OUP 1995), henceforth referred to as ABC. While some of them are new (i.e. not discovered at the time of publication of ABC), others were deliberately left out of ABC because they were not thought significant enough, or because they did not fit the format of ABC; but we believe that they are worth keeping on record.

As in ABC, we denote mathematical errors by *** and new information by NEW. We use C to denote improvements concerning grammar or notational consistency; and M to refer to misprints or cases of illegibility. Changes are shown in order of appearance. New references are not shown. The inclusion or otherwise of any modification is not intended to imply anything about its inclusion in any future ATLAS publication.

A character table in ATLAS format has been obtained for ^{2}E_{6}(2).3.

Introduction

C Page iv  : Top, insert “page” before “number”.
M Page vii : Top left, change head of Section 1 to “Preliminaries”.
C  : Section 2, change “to” to “from” after “differently”.
C Page viii : Table 1, order of O′N, interchange 5 and 7^{3}.

NEW Page xi : Middle right, for sake of completeness append description of orthogonal groups in

dimensions 1 and 2.

C Page xiii : Section 6, 2nd paragraph begins: “In the Clifford algebra, the vectors…generate a

multiplicative group which is…”.

C Page xiv : Middle left, omit “to” in “also to specify”.
C Page xx : Middle right, insert “it” between “which” and “becomes”.

*** Page xxiii : Section 7, in middle of 4th paragraph, insert “irreducible” between “faithful” and

“representation”.

M Page xxv : Section 5, insert i in “contans”.
C Page xxvi : In character table of 2A_{5}, replace −0 by 0.
C Page xxix : Middle right, singularize “act”.

NEW : In Section 18, specify notational convention where both coset and cohort are

“squashed”.

M Page xxxi : Section 1, replace “product” by “produce”.

The Groups

NEW Page 4 A_{6}  : In “Linear” section replace in 2nd line G.2 by G.2_{2}.
C Page 12 L_{2}(16) : Singularize “Presentations”.
C Page 14 U_{3}(3) : In 3rd Max, replace 4.S_{4} :2 by 2_{+}^{1+4}.S_{3}.
*** Page 16 L_{2}(25) : In 5th Max, replace Q_{8} × S_{3} by (Q_{8} × 3).2.
*** Page 17  : Change element orders of 4C, 12C, 12D in 4.G.2_{3} to 4, 12, 12.
C Page 21 L_{2}(31) : Singularize “Presentations”.
C Page 24 L_{3}(4) : Remove header line.

*** Page 28 Sz(8) : In “Suzuki” construction, insert before comma in 1st line “and such that when

t = 0 we have x = y = 0”.

C Page 29 L_{2}(32) : Singularize “Presentations”.
NEW Page 30 U_{3}(4) : In 3rd Max, N(5^{2}) is also N(5A–D_{3}EF_{3}).
***  : Change class names 10A–B in G.2 to 10E–F.
C Page 31 M_{12}  : In MINIMOG section close bracket in 6th line.
NEW Page 33  : In 9th Max, replace H.2 by M_{8}.(S_{4} × 2).
M Page 36 J_{1}  : Raise most of bottom line in “Graph” section.
C  : Replace = by ≅ in presentation.

C  : In 6th Max, replace D_{6} by S_{3}.
NEW Page 42 J_{2}  : In 4th Max, replace H.2 by 2^{2+4}.(S_{3} × S_{3}).
*** Page 52 U_{4}(3) : In 6th Max, replace 3_{+}^{1+4}.(2S_{4} × 2) by 3_{+}^{1+4}.4S4.
NEW Page 53  : 1st Max is N(3^{4}) = N(3A_{10}B_{15}C_{15}).
*** Page 55  : Change class name 12J in G.2_{3} to 12K. This also applies on page 57 and 59.

*** Page 60 G_{2}(3) : In “14-dimensional” construction, in 5th line image of e_{i} should be divided

by 3.

C Page 66 U_{3}(8) : Remove the note from the map that had been suggested in Appendix 2 of

ABC, since the isoclines of 3.G.3_{1} are in fact isomorphic.

C Page 68 L_{4}(3) : Swap the two fusion markers for χ_{31}–χ_{34} in 2.G.2_{1}.
NEW Page 70 L_{5}(2) : In 3rd and 4th Maxes, N(2^{6}) is also N(2A_{21}B_{42}).
C Page 73 U_{5}(2) : In 4th Max, remove brackets round AB and CD.
NEW Page 74 L_{3}(8) : In 3rd Max, N(7^{2}) is also N(7A–F_{3}GH_{2}IJK_{3}).

*** Page 77 Sz(32) : In “Suzuki” construction, insert before comma in 2nd line “and such that

when t = 0 we have x = y = 0”.

NEW Page 79 U_{3}(9) : In 4th Max, N(5^{2}) is also N(5A–D_{3}EF_{3}).

C Page 80 HS : 2nd, 3rd, 6th, 7th, 9th and 11th Maxes respectively fix 233−2323, 233−2332, 233 − 2433, 233 − 2433, 233 − 2343 and 233 − 2433 points. For the 6th, 7th and 11th this is a further change, in all cases designed to conform with a convention that in abc − defg the sums of (the vector corresponding to) d with a, b, c are e, f, g respectively.

C Page 81  : Replace ∗7 by ∗2 in χ_{42}.

*** Page 82 J_{3} : In “Unitary” construction, the definition of C_{Z} only applies when z ≠ 0: the

image of e(0) is e(0).

C Page 83  : Replace r6 ∗ by r6 −r6 in χ_{9}.
C Page 89 O_{8}^{−}(2) : Pluralize “Presentation”.

*** ^{3}D_{4}(2) : In “Jordan” construction, replace top row (but not 1st column) occurrences

of \frac{1}{4}s by \frac{1}{4} \overline{s} in 768 matrix orbit.

M  : In line after display of roots in same section, delete comma before slash.

NEW : New presentation for G is linear Coxeter graph ⟨a6b3c3d | ((ab)^{2}c)^{6}, (a^{bc}(a^{b^{c}b^{a}}a^{bcd})^{2})^{3}, (abcd)^{21}⟩. Omitting the last relator gives a presentation for S_{4} × G.

NEW  : In 7th Max, N(7^{2}) is also N(7A–C_{4}D_{4}).
*** Page 97 G_{2}(4) : In 2nd Max, delete entry in last column.

C Page 100 McL : 3rd, 7th and 10th Maxes respectively fix 223−2324, 223−2443 and 223−2434

points. For the 7th this is a further change. See amendment to Page 80 above.

NEW Page 104 He  : 5th Max is split.
C Page 109 O_{7}(3) : Pluralize “Presentation”.
***  : In 14th Max, replace 2E by 2F.
C Page 113 S_{6}(3) : Move “Specifications” to right.
NEW  : 2nd Max is N(3^{6}) = N(3AB_{13}C_{39}D_{78}E_{234}).
NEW  : 3rd Max is N(3^{3}) = N(3AB_{4}C_{3}D_{6}).
***  : In 7th Max, extensions are split.
NEW Page 115 U_{6}(2) : N(2^{9}) is also N(2A_{21}B_{210}C_{280}).

C : 5th, 6th, 11th and 12th Maxes respectively fix 222 − 2323, 222 − 2332, 222 −

2433 and 222 − 2343 points. See amendment to Page 80 above.

*** Page 123 R(27) : In 1st Max, replace both occurrences of 3^{3+6} by 3^{3+3+3}.
C  : Insert separator between R(27) and S_{8}(2).
C  S_{8}(2) : Pluralize “a presentation”.
NEW  : 7th Max is N(2^{6}) = N(2A_{7}B_{7}C_{21}E_{28}).
C Page 129 Suz  : Replace r10 ∗ by r10 −r10 in χ_{40}.
C Page 131  : Pluralize “Presentation”.

***  : 12th Max is non-split.
C Page 140 O_{8}^{+}(3) : Pluralize “a presentation”.
NEW  : Maxes 7-9 are also N(3^{6}).
***  : In 8th and 9th Maxes, replace L_{4}(3) × 2 by L_{4}(3):2_{1}.
NEW Page 146 O_{10}^{+}(2) : In 2nd Max, N(2^{8}) is also N(2A_{135}B_{120}).
NEW  : In 3rd and 4th Maxes, N(2^{10}) is also N(2A_{155}C_{868}).
NEW Page 147 O_{10}^{−}(2) : In 1st Max, N(2^{8}) is also N(2A_{119}B_{136}).
NEW  : In 5th Max, N(2^{6}) is also N(2A_{35}C_{28}).
NEW  : In 6th Max, replace N(2^{3}) by N(2A^{3}).
C  : In last Max, N(3A_{2}D_{2}) is also N(3^{2}).
NEW Page 154 Co_{2}  : In 2nd Max, N(2^{10}) is also N(2A_{77}B_{330}C_{616}).

*** Page 160 F i_{22} : In the additional five columns, on p. 310 of ABC, the centralizer orders of

12L, 12M and 12N are 3456, 1296 and 576 respectively.

*** Page 163  : 7th Max is (2 × 2_{+}^{1+8}):U_{4}(2) : 2 extending to H:2.
NEW  : In 8th Max, replace N(2^{4}) by N(2B^{4}).
C Page 172 F_{4}(2) : Replace −0 by 0 in χ_{96} on 10A.
NEW Page 177 F i_{23}  : In 6th Max, N(2^{11}) is also N(2A_{23}B_{253}C_{1771}).
NEW  : In 11th Max, N(2^{6}) is also N(2A_{7}B_{21}C_{35}).
C Page 182 Co_{1}  : Near bottom, pluralize “portion”, “group” and “parenthesis”.
*** Page 183  : Maximal subgroup 5^{3} : (4 × A_{5}).2 is in fact 5^{3} : (4 × S_{5}).
*** Page 191 ^{2}E_{6}(2) : 16th Max is 3^{1+6} :2^{3+6} :3^{2} :2.
C  : In last Max, replace 3^{2}Q_{8} by 3^{2} :Q_{8}.

NEW Page 217 B : New Maxes M_{11}, L_{3}(3), L_{2}(17) : 2 and L_{2}(11) : 2. List is now complete. Furthermore, in the group (S_{6} × L_{3}(4) : 2) : 2 quoted in the original list of amendments, the 1st “:2” is in fact “:2_{2}”.

C Page 230 M  : The symbol 𝕄 is now used for the Monster.
C  : Pluralize “Presentation”.

NEW : New presentation for G has generators s, t, v, x and relators s^{6}, t^{3}, (st)^{4}, (s^{2}t)^{4}, (s^{3}t)^{3}, [s^{2}, (ts^{2}t)^{2}], [v, ut^{−1}], [v, u^{3}su^{−2}], v^{2}, [v, v^{t}], (vu)^{13}, [vx, ut^{−1}], [vx, s^{u^{3}}], x^{3}, (v^{u}vx)^{2}, (x^{−1}x^{s})^{2}, (xt^{−1})^{12}, (u^{−6}xu^{6}s)^{6}(sux^{−1}u^{−1})^{6}s^{−1} and ((xv^{u^{4}}v^{u^{10}})^{3}u)^{13}. s and t are the generators for L_{3}(3) used on page 13, and their permutation action on the points of the projective plane can be written as s = (1, 2, 5, 9, 8, 7)(3, 12, 4)(10, 11) and t = (0, 12, 3)(1, 2, 4)(5, 7, 11)(6, 9, 8). v is the product of all points of the projective plane except the one corresponding to 0, and x is the product of that point with the line corresponding to (0, 1, 3, 9). Omitting the penultimate or last relators, or both, gives the groups G × L_{3}(3), G × G, and G × G × L_{3}(3) respectively.

M  : Replace , by . in last line before list of groups.

NEW Page 231 : A list of groups which may be involved or contained in the Monster has been

compiled. See below.

NEW Page 234 : The list of Maxes that are 2-local or normalizers of products of two or more simple groups is complete. New Maxes L_{2}(19) : 2, L_{2}(29) : 2, L_{2}(41), L_{2}(59) and L_{2}(71) – the last two implying that 59.29 and 71.35 are not maximal.

Additional Information and Bibliography

C Page 239 : Pluralize “Order”.

NEW : Insert sentence: groups whose full character tables are in the ATLAS are starred, while a bracketed star is appended to other ATLAS groups and to those for which all or part of the character table exists in ATLAS format. Also, insert relevant stars, with or without brackets, in pages 239-242.

*** Page 241 : Outer automorphism group of G_{2}(9) is 4.
M Page 247 : line 3: Replace 243 by 323.

Which simple groups are contained or involved in the Monster?

We divide all simple groups whose order divides that of 𝕄 into the following four types:

(a) The following groups are subgroups of 𝕄 and their occurrences are completely classified: C_{p} (p prime dividing |𝕄|), A_{n} (5 ≤ n ≤ 12), L_{2}(q) (q = 7, 8, 11, 17, 19, 23, 25, 29, 31, 41, 49, 59, 71), L_{4}(3), U_{3}(5), U_{5}(2), S_{4}(4), S_{n}(2) (n = 6, 8), O_{7}(3), O_{8}^{+}(q) (q = 2, 3), O_{n}^{−}(2) (n = 8, 10), G_{2}(3), ^{3}D_{4}(2), ^{2}F_{4}(2)′, M_{12}, F i_{23}, He, HN.

(b) The following groups are subgroups of 𝕄 but their occurrences have not been completely classified: L_{2}(q) (q = 13, 16).

(c) The following groups are known to be involved in 𝕄 but not contained: L_{2}(81), L_{3}(q) (q = 4, 5), L_{5}(2), U_{4}(3), U_{6}(2), O_{8}^{−}(3), O_{10}^{+}(2), G_{2}(4), F_{4}(2), ^{2}E_{6}(2), M_{n} (n = 22, 23, 24), Co_{n} (n = 1, 2, 3), F i_{22}, F i′_{24}, J_{2}, Suz, HS, McL, B.

(d) The following groups are not involved in 𝕄: A_{n} (13 ≤ n ≤ 32), L_{2}(q) (q = 27, 32, 47, 64, 125, 169, 1024), L_{3}(q) (q = 7, 9, 16, 25), L_{4}(q) (q = 4, 5, 7, 9), L_{5}(q) (q = 3, 4), L_{6}(q) (q = 2, 3, 4), U_{4}(q) (q = 4, 5, 8), U_{n}(4) (n = 5, 6), S_{4}(q) (q = 5, 7, 8, 9), S_{6}(q) (q = 3, 4, 5), S_{8}(3), S_{n}(2) (n = 10, 12), O_{7}(5), O_{9}(3), O_{10}^{+}(3), O_{12}^{±}(2), Sz(8), Sz(32), G_{2}(5), J_{n} (n = 1, 3), O′N, Ru.

The number of changes of each type is as follows: 22*** 32NEW 39C 7M.

Last update 16 March 2017; this file has 231 lines.