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 2E6(2).3.


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 73.
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 2A5, 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   A6   :   
In "Linear" section replace in 2nd line G.2 by G.22.
C   Page   12   L2(16)   :   
Singularize "Presentations".
C   Page   14   U3(3)   :   
In 3rd Max, replace 4.S4:2 by 2+1+4.S3.
***   Page   16   L2(25)   :   
In 5th Max, replace Q8×S3 by (Q8×3).2.
***   Page   17      :   
Change element orders of 4C, 12C, 12D in 4.G.23 to 4, 12, 12.
C   Page   21   L2(31)   :   
Singularize "Presentations".
C   Page   24   L3(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   L2(32)   :   
Singularize "Presentations".
NEW   Page   30   U3(4)   :   
In 3rd Max, N(52) is also N(5A-D3EF3).
***            :   
Change class names 10A-B in G.2 to 10E-F.
C   Page   31   M12   :   
In MINIMOG section close bracket in 6th line.
NEW   Page   33      :   
In 9th Max, replace H.2 by M8.(S4×2).
M   Page   36   J1   :   
Raise most of bottom line in "Graph" section.
C            :   
Replace = by ≅ in presentation.
C            :   
In 6th Max, replace D6 by S3.
NEW   Page   42   J2   :   
In 4th Max, replace H.2 by 22+4.(S3×S3).
***   Page   52   U4(3)   :   
In 6th Max, replace 31+4+.(2S4 ×2) by 31+4+.4S4.
NEW   Page   53      :   
1st Max is N(34)=N(3A10B15C15).
***   Page   55      :   
Change class name 12J in G.23 to 12K. This also applies on page 57 and 59.
***   Page   60   G2(3)   :   
In "14-dimensional" construction, in 5th line image of ei should be divided by 3.
C   Page   66   U3(8)   :   
Remove the note from the map that had been suggested in Appendix 2 of ABC, since the isoclines of 3.G.31 are in fact isomorphic.
C   Page   68   L4(3)   :   
Swap the two fusion markers for χ3134 in 2.G.21.
NEW   Page   70   L5(2)   :   
In 3rd and 4th Maxes, N(26) is also N(2A21B42).
C   Page   73   U5(2)   :   
In 4th Max, remove brackets round AB and CD.
NEW   Page   74   L3(8)   :   
In 3rd Max, N(72) is also N(7A-F3GH2IJK3).
***   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   U3(9)   :   
In 4th Max, N(52) is also N(5A-D3EF3).
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 χ42.
***   Page   82   J3   :   
In "Unitary" construction, the definition of CZ only applies when z ≠ 0: the image of e(0) is e(0).
C   Page   83      :   
Replace r6 * by r6 −r6 in χ9.
C   Page   89   O8(2)   :   
Pluralize "Presentation".
***         3D4(2)   :   
In "Jordan" construction, replace top row (but not 1st column) occurrences of ¼ s by ¼ 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)2c)6,(abc(abcbaabcd)2)3,(abcd)21〉. Omitting the last relator gives a presentation for S4×G.
NEW            :   
In 7th Max, N(72) is also N(7A-C4D4).
***   Page   97   G2(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   O7(3)   :   
Pluralize "Presentation".
***            :   
In 14th Max, replace 2E by 2F.
C   Page   113   S6(3)   :   
Move "Specifications" to right.
NEW            :   
2nd Max is N(36)=N(3AB13C39D78E234).
NEW            :   
3rd Max is N(33)=N(3AB4C3D6).
***            :   
In 7th Max, extensions are split.
NEW   Page   115   U6(2)   :   
N(29) is also N(2A21B210C280).
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 33+6 by 33+3+3.
C            :   
Insert separator between R(27) and S8(2).
C         S8(2)   :   
Pluralize "a presentation".
NEW            :   
7th Max is N(26)=N(2A7B7C21E28).
C   Page   129   Suz   :   
Replace r10 * by r10 −r10 in χ40.
C   Page   131      :   
Pluralize "Presentation".
***            :   
12th Max is non-split.
C   Page   140   O8+(3)   :   
Pluralize "a presentation".
NEW            :   
Maxes 7-9 are also N(36).
***            :   
In 8th and 9th Maxes, replace L4(3)×2 by L4(3):21.
NEW   Page   146   O10+(2)   :   
In 2nd Max, N(28) is also N(2A135B120).
NEW            :   
In 3rd and 4th Maxes, N(210) is also N(2A155C868).
NEW   Page   147   O10(2)   :   
In 1st Max, N(28) is also N(2A119B136).
NEW            :   
In 5th Max, N(26) is also N(2A35C28).
NEW            :   
In 6th Max, replace N(23) by N(2A3).
C            :   
In last Max, N(3A2D2) is also N(32).
NEW   Page   154   Co2   :   
In 2nd Max, N(210) is also N(2A77B330C616).
***   Page   160   Fi22   :   
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):U4(2):2 extending to H:2.
NEW            :   
In 8th Max, replace N(24) by N(2B4).
C   Page   172   F4(2)   :   
Replace −0 by 0 in χ96 on 10A.
NEW   Page   177   Fi23   :   
In 6th Max, N(211) is also N(2A23B253C1771).
NEW            :   
In 11th Max, N(26) is also N(2A7B21C35).
C   Page   182   Co1   :   
Near bottom, pluralize "portion", "group" and "parenthesis".
***   Page   183      :   
Maximal subgroup 53:(4 ×A5).2 is in fact 53:(4 ×S5).
***   Page   191   2E6(2)   :   
16th Max is 31+6:23+6:32:2.
C            :   
In last Max, replace 32Q8 by 32:Q8.
NEW   Page   217   B   :   
New Maxes M11, L3(3), L2(17):2 and L2(11):2. List is now complete. Furthermore, in the group (S6×L3(4):2):2 quoted in the original list of amendments, the 1st ":2" is in fact ":22".
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 s6, t3, (st)4, (s2t)4, (s3t)3, [s2,(ts2t)2], [v,ut−1], [v,u3su−2], v2, [v,vt], (vu)13, [vx,ut−1], [vx,su3], x3, (vuvx)2, (x−1xs)2, (xt−1)12, (u−6xu6s)6(sux−1u−1)6s−1 and ((xvu4vu10)3u)13. s and t are the generators for L3(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×L3(3), G×G, and G×G×L3(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 L2(19):2, L2(29):2, L2(41), L2(59) and L2(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 G2(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: Cp (p prime dividing |𝕄 |), An (5 ≤ n ≤ 12), L2(q) (q=7,8,11,17,19,23,25,29,31,41,49,59,71), L4(3), U3(5), U5(2), S4(4), Sn(2) (n=6,8), O7(3), O8+(q) (q=2,3), On(2) (n=8,10), G2(3), 3D4(2), 2F4(2)′, M12, Fi23, He, HN.

(b) The following groups are subgroups of 𝕄 but their occurrences have not been completely classified: L2(q) (q=13,16).

(c) The following groups are known to be involved in 𝕄 but not contained: L2(81), L3(q) (q=4,5), L5(2), U4(3), U6(2), O8(3), O10+(2), G2(4), F4(2), 2E6(2), Mn (n=22,23,24), Con (n=1,2,3), Fi22, Fi24′, J2, Suz, HS, McL, B.

(d) The following groups are not involved in 𝕄: An (13 ≤ n ≤ 32), L2(q) (q=27,32,47,64,125,169,1024), L3(q) (q=7,9,16,25), L4(q) (q=4,5,7,9), L5(q) (q=3,4), L6(q) (q=2,3,4), U4(q) (q=4,5,8), Un(4) (n=5,6), S4(q) (q=5,7,8,9), S6(q) (q=3,4,5), S8(3), Sn(2) (n=10,12), O7(5), O9(3), O10+(3), O12±(2), Sz(8), Sz(32), G2(5), Jn (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.

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On 16 Mar 2017, 20:27.