Quelle atlasimp.json Sprache: unbekannt

{
  "version": "Jan 06, 2021",
  "description": "the contents of the lists of Improvements to the ATLAS of Finite Groups, extended by a tagging of the errors (***) with I, CH, G, MX, FI, P, MP; the 'source' values are 1 for entries from [BN95], and 2 for entries from [Nor]",
  "data": [
  {
   "source": "1",
   "section": "Introduction",
   "entries": [
[ "M", "p. viii", "", "Top right, replace $F_{24}$ by $Fi_{24}$." ],
[ "M", "p. x", "", "Replace {\\sl F} in 2nd line of Section 2 by $\\FF$." ],
[ "***", "p. xi", "", "Middle left, replace $C_n(q)$ by $C_m(q)$; top right, replace $L_n(q)$ by $GL_n(q)$.", "I" ],
[ "M", "p. xi", "", "Bottom right, insert `$(q)$' in symbol defined by Dieudonn\\'e." ],
[ "***", "p. xii", "", "Middle left, replace $n$, $2n-2$ by $m$, $2m-2$ in formula for N.", "I" ],
[ "NEW", "p. xv", "", "Section 3, replace $d,f,g$ by $d,e,f,g$." ],
[ "M", "p. xv", "", "Section 5, before `the exceptional isomorphisms' insert `the identity of $B_n$ and $C_n$ in characteristic $2$ and'." ],
[ "M", "p. xv", "", "Section 6, replace `the' by `a' before `Chevalley group determined'." ],
[ "M", "p. xvi", "", "Table 6, in formulae for $B_n(q)$, ${}^2A_n(q)$, ${}^2D_n(q)$ replace $t$ by $i$." ],
[ "***", "p. xvi", "", "Table 7, line beginning $A_n$, replace final $n-1$ by $n+1$. Line beginning $D_n$, replace each $2^n$ by $2^{n-1}$.", "I" ],
[ "NEW", "p. xx", "", "Insert the following sentence just before the solid line:<<nl>>Note that our notation does not specify a unique group in all cases; and in cases of ambiguity different occurrences of the same notation may correspond to different groups." ],
[ "M", "p. xxiv", "", "Guide, in references to \\S\\S 7, 8, 16, 18, replace upper dots by lower dots; furthermore in references to \\S\\S 13, 16 replace $M$ by $m$, $a$ respectively." ],
[ "M", "p. xxv", "", "Section 3, replace 15 by 16 in parenthetical reference." ],
[ "NEW", "p. xxviii", "", "Add to Section 13:<<nl>>The {\\sl Schur index} of $\\chi$ over $H$ may be defined as follows. Consider the field obtained by adjoining to $\\QQ$ the values of $\\chi$. Any characteristic zero representation of $H$ affording $\\chi$ will be written over an extension of the above field. The least degree of such an extension is the Schur index of $\\chi$.<<nl>>For almost all characters occurring in the {\\sc Atlas}, the Schur index is either $1$ or $2$. Furthermore, the latter tends to occur only for symplectic characters (those with indicator $-1$), whose Schur index is always even. If $G$ is a sporadic group, then the only exceptions to the above statements for extension groups of type $m.G.a$ are non-symplectic characters of Schur index $2$, and these are all listed in the `remarks' section under the corresponding simple group $G$." ],
[ "***", "p. xxix", "", "In $2.A_5.2$ table, $\\chi_{6,7}$ is $2$ on $3A_1$.", "CH" ],
[ "***", "p. xxix", "", "Last line ends `divisor of $N$ prime to $n$'.", "I" ],
[ "***", "p. xxx", "", "In line 5, replace $n$ by $N$.", "I" ],
[ "NEW", "p. xxx", "", "Insert paragraph at end of Section 19:<<nl>>It should be noted that these conventions may be counter-intuitive. For example, if the printed values of a character of a triple cover involve a single irrationality, then the values of its proxy on the same classes are equal to those of the character itself if the irrationality is $b5$, but to those of the conjugate printed character if the irrationality is $b15$." ],
[ "NEW", "p. xxx", "", "In Section 20, insert that fusion symbols in abbreviated tables are as in Section 18." ],
[ "***", "p. xxxii", "", "Middle left, delete $1/|G|$ in formula for $R(g)$.", "I" ],
[ "M", "p. xxxii", "", "Middle Right, hyphenate Zia ud Din and replace D. Enomoto by H. Enomoto." ]
   ]
  },
  {
   "source": "1",
   "section": "The Groups",
   "entries": [
[ "NEW", "p. 7", "$L_2(11)$", "New presentation<<nl>>$\\langle a,b,c\\mid a^2=b^2=c^2=(ab)^3=(ac)^3=(bc)^6=(abcbc)^3=1\\rangle$." ],
[ "NEW", "p. 13", "$L_3(3)$", "New presentation for $G{:}2$ is<<nl>>$\\langle \\begin{picture}(140,15)(-10,10) \\put(0,10){\\line(1,0){120}} \\put(0,10){\\circle*{4}} \\put(30,10){\\circle*{4}} \\put(60,10){\\circle*{4}} \\put(90,10){\\circle*{4}} \\put(120,10){\\circle*{4}} \\put(-2,15){$a$}\\put(28,15){$b$}\\put(58,15){$c$}\\put(88,15){$d$}\\put(118,15){$e$} \\put(43,15){6}\\put(73,15){8} \\end{picture}<<nl>>\\mid a=(cd)^4,e=(bc)^3\\rangle $." ],
[ "***", "p. 14", "$U_3(3)$", "In Cayley $(2)$ section, isotropic elements square to themselves, not $0$.", "G" ],
[ "***", "p. 14", "$U_3(3)$", "In Cayley $\\ZZ$ section, insert ${1\\over 2}$ after $=$ in line beginning `Alternatively,'. \\vspace{1mm}", "G" ],
[ "NEW", "p. 16", "$L_2(25)$", "New presentations $\\langle a,b\\mid a^2=b^3=(ab)^{13}=((ab)^3(ab^{-1})^4)^2=1\\rangle$ for $G$ and<<nl>>$\\langle \\begin{picture}(110,15)(-10,10) \\put(0,10){\\line(1,0){90}} \\put(0,10){\\circle*{4}} \\put(30,10){\\circle*{4}} \\put(60,10){\\circle*{4}} \\put(90,10){\\circle*{4}}\\put(-2,15){$a$}\\put(28,15){$b$}\\put(58,15){$c$}\\put(88,15){$d$}\\put(43,15){6} \\end{picture}<<nl>>\\mid (abc)^4=(bcd)^{10}=1\\rangle$ for $2\\times G{:}2_2$." ],
[ "***", "p. 17", "$L_2(25)$", "In $\\chi_{14}, \\chi_{15}$ change sign of $r3$.", "CH" ],
[ "NEW", "p. 18", "$M_{11}$", "New presentation: to that given above for $L_2(11)$ add involution $d$ with $(ad)^2=(cd)^2=(bcbcd)^2=(bd)^3=1$." ],
[ "NEW", "p. 18", "$L_2(27)$", "New presentation $\\langle a,b\\mid a^2=b^3=(ab)^{13}=((ab)^3(ab^{-1})^3)^2=1\\rangle$." ],
[ "***", "p. 19", "$L_2(27)$", "Conjugate character values on $9A, 9B$.", "CH" ],
[ "***", "p. 21", "$L_2(31)$", "Insert two new maximal subgroups $S_4$ interchanged by the outer automorphism, with order 24, index 620, Abstract Specification $N(2A^2)$ and Orthogonal Specification `base'.", "MX" ],
[ "***", "p. 25", "$L_3(4)$", "Conjugate $\\chi_{35}, \\chi_{40}$ on $3B,15AB,6C$.", "CH" ],
[ "***", "p. 30", "$U_3(4)$", "Join $\\chi_{7}, \\chi_{8}$ in fusion column of $G.4$.", "FI" ],
[ "***", "p. 32", "$M_{12}$", "In Hadamard section line 4, replace $i$ by $i - j$.", "G" ],
[ "NEW", "p. 32", "$M_{12}$", "New presentation (for $2.G$): to that given above for $M_{11}$ add involution $e$ with $(ae)^2=(be)^2=(de)^2=(bcbce)^2=(ce)^3=1$." ],
[ "NEW", "p. 32", "$M_{12}$", "Insert remarks that $M_{12}$ is contained in $E_6(5)$, and that $\\chi_{20,21}$ has Schur index $2$ over $2.G.2$." ],
[ "***", "p. 34", "$U_3(5)$", "In Graph section, replace $A_7$ (twice) by $S_7$.", "G" ],
[ "NEW", "p. 34", "$U_3(5)$", "Presentation for $G.2$ is<<nl>>$\\langle S_7,a\\mid a^2=[a,(12345)]=(14)(23)(a(56))^2=(a(67))^5=1\\rangle$." ],
[ "NEW", "p. 34", "$U_3(5)$", "In 4th maximal subgroup, insert `bisection' in graph specification." ],
[ "NEW", "p. 36", "$J_1$", "In 2nd maximal subgroup, insert `set of $7$ edges' in graph specification." ],
[ "***", "p. 38", "$L_3(5)$", "Conjugate $\\chi_{4}, \\chi_5, \\chi_{28}, \\chi_{29}$ on $8A$--$8B$, $24A$--$D$ and $\\chi_{22}$--$\\chi_{25}$ on $4AB$, $12AB$, $20AB$.", "CH" ],
[ "***", "p. 39", "$M_{22}$", "In Unitary section, replace `Each non-zero isotropic vector' by `There is an orbit of 231 isotropic 1-spaces, each of which'.", "G" ],
[ "NEW", "p. 39", "$M_{22}$", "New presentation (for $2.G$): to that given above for $M_{11}$ add relation $(abcdcb)^4=1$ and replace $(cd)^2=1$ by $(cd)^3=1$." ],
[ "NEW", "p. 39", "$M_{22}$", "Insert remark that the following characters have Schur index $2$: $\\chi_{30}$ and its conjugate over $4.G$; the fusion of these two over the group of type $4.G.2$ {\\sl isoclinic to} the one shown; the fusion of a $\\chi_{26}$ and the corresponding $\\chi_{27}$ over the same group; and the fusion of $\\chi_{52}$ and its conjugate over the group of type $6.G.2$ shown." ],
[ "***", "p. 42", "$J_2$", "In Quaternionic section, interchange $192$ and $96$.", "G" ],
[ "NEW", "p. 42", "$J_2$", "Add remark that $\\chi_{21}$ has Schur index 2 over $G$." ],
[ "NEW", "p. 42", "$J_2$", "In 6th maximal subgroup, insert `set of $10$ decads' in graph specification." ],
[ "***", "p. 46", "$S_6(2)$", "Replace $2B$ by $2D$ in specification of last maximal subgroup.", "MX" ],
[ "NEW", "p. 50", "$L_3(7)$", "Presentation for $G{:}2$ is<<nl>>$\\langle \\begin{picture}(170,15)(-10,10) \\put(0,10){\\line(1,0){150}} \\put(0,10){\\circle*{4}} \\put(30,10){\\circle*{4}} \\put(60,10){\\circle*{4}} \\put(90,10){\\circle*{4}} \\put(120,10){\\circle*{4}} \\put(150,10){\\circle*{4}} \\put(-2,15){$a$}\\put(28,15){$b$}\\put(58,15){$c$}\\put(88,15){$d$} \\put(118,15){$e$}\\put(148,15){$f$}\\put(73,15){8} \\end{picture} \\mid af=(cd)^4\\rangle$." ],
[ "***", "p. 51", "$L_3(7)$", "$16F$ squares to $8B$, not $8A$.", "P" ],
[ "***", "p. 51", "$L_3(7)$", "Change sign of $r2$ in $\\chi_{14}, \\chi_{15}$ on $16EF$ and $\\chi_{35}, \\chi_{36}$ on $48A$--$D$.", "CH" ],
[ "***", "p. 52", "$U_4(3)$", "In 1st line of Reflection section, insert $w$ after $+$.", "G" ],
[ "***", "p. 52", "$U_4(3)$", "In 1st maximal subgroup, in $G.D_8$ column, replace $(2\\times A_6.2^2)$ by $(2\\times A_6).2^2$.", "MX" ],
[ "***", "p. 52", "$U_4(3)$", "In 8th and 9th maximal subgroups, in $G.2_1$ column replace $2^4{:}S_6$ by $H.2$.", "MX" ],
[ "***", "p. 52", "$U_4(3)$", "In 14th maximal subgroup, replace $.4$ by $.2^2$ (three times).", "MX" ],
[ "***", "p. 53", "$U_4(3)$", "In map, the rectangles corresponding to automorphisms of type $2_2$ or $2_3$ acting on triple or sextuple covers, where just one of the cover and automorphism is dashed, should be open at the right and bottom.", "MP" ],
[ "NEW", "p. 53", "$U_4(3)$", "1st maximal subgroup is $N(3^4)=N(3A_{10}B_{15}C_{15})$." ],
[ "NEW", "p. 53", "$U_4(3)$", "8th and 9th maximal subgroups are both $N(2A^4)$." ],
[ "***", "p. 55", "$U_4(3)$", "Change sign of $6i3$ wherever it occurs in the $6HI$ columns on this page. Also $18A$ and $18B$ cube to $6H$, $6I$ respectively.", "CH" ],
[ "***", "p. 55", "$U_4(3)$", "In $4.G.2_2$ and $4.G.2_3$ orders of $4I,12J,8H$ should be doubled.", "P" ],
[ "***", "p. 55", "$U_4(3)$", "In $4.G.4$, $\\chi_{48}$ has indicator $\\circ\\!\\circ\\!2$ not $\\circ 2$, and fuses to $\\chi_{49}$ not $\\chi_{47}$.", "FI" ],
[ "***", "p. 56", "$U_4(3)$", "In last lifting order row of $12_1.G$, `$24\\ 84\\ 8$' should be `$84\\ 84\\ 24$'.", "P" ],
[ "***", "p. 57", "$U_4(3)$", "In $12_1.G.2_2$ and $12'_1.G.2_2$ orders of $4I$ and $12J$ should be doubled.", "P" ],
[ "***", "p. 59", "$U_4(3)$", "In $6_2.G.2_3$ and $12_2.G.2_3$, $6N$ splits into two classes, and in latter group, also $12'_2.G.2_3$, order of $8H$ should be doubled.", "P" ],
[ "***", "p. 60", "$G_2(3)$", "In $14$-dimensional section, replace first occurrence of $5i+4$ in line 4 by $5i+6$; also interchange of two orbits shown is up to scalar multiplication.", "G" ],
[ "NEW", "p. 60", "$G_2(3)$", "Presentation for $G$ can be obtained from that for $L_3(3){:}2$ by adjoining generator $f$ joined to $a,b,c$ and relations $(abf)^2=(cbfcde)^3=1$." ],
[ "NEW", "p. 61", "$S_4(5)$", "Presentation for $2\\times G{:}2$ is<<nl>>$\\langle\\begin{picture}(140,15)(-10,10) \\put(0,10){\\line(1,0){120}} \\put(0,10){\\circle*{4}} \\put(30,10){\\circle*{4}} \\put(60,10){\\circle*{4}} \\put(90,10){\\circle*{4}} \\put(120,10){\\circle*{4}} \\put(-2,15){$a$}\\put(28,15){$b$}\\put(58,15){$c$}\\put(88,15){$d$}\\put(118,15){$e$} \\put(43,15){6}\\put(73,15){5} \\end{picture} \\mid (abc)^4=(bcd)^{10}=1\\rangle$. Also, Abstract Specification of second maximal subgroup is $N(5^3)= N(5AB_6C_{10}D_{15})$." ],
[ "***", "p. 64", "$U_3(8)$", "In $3.G.3_1$ orders of $3EF,6DE,12BC$ should be tripled.", "P" ],
[ "***", "p. 64", "$U_3(8)$", "$\\chi_{33}, \\chi_{34}$ should be $2z9,2z9\\!*\\!*$ on $12B$ and $12C$ respectively.", "CH" ],
[ "***", "p. 64", "$U_3(8)$", "In $3.G.3_2$ values of $\\chi_{29}$--$\\chi_{49}$ should be multiplied by $z9$. See pp.~\\pageref{U38a}--\\pageref{U38b} for a possible version of the table in {\\sc Atlas} form.", "CH" ],
[ "***", "p. 65", "$U_3(8)$", "$24B$ cubes to $8B$, not $8A$.", "P" ],
[ "NEW", "p. 66", "$U_3(8)$", "Insert note in map:<<nl>>There is a unique group of type $3.G.6$ which contains the group of type $3.G.3$ shown. But the (unique) groups of type $3.G.6'$ and $3.G.6''$ contain not this $3.G.3$ but its {\\sl isoclines}.<<nl>>NO, do NOT insert this note, see the 2nd part of the improvements." ],
[ "***", "p. 67", "$U_3(7)$", "$3'$ part of $12B$ is $4B$, not $4A$.", "P" ],
[ "***", "p. 67", "$U_3(7)$", "$7'$ parts of $14A,28A,28B$ are $2A,4A,4B$.", "P" ],
[ "***", "p. 68", "$L_4(3)$", "Indicators of $\\chi_{23-26}$ are $\\circ$ in $G$.", "FI" ],
[ "***", "p. 69", "$L_4(3)$", "In 3rd maximal subgroup, replace $2.(S_4\\times S_4).2\\times 2$ by $2.(S_4\\times S_4\\times 2).2$.", "MX" ],
[ "***", "p. 69", "$L_4(3)$", "In 7th maximal subgroup, replace $S_6$ by $PGL_2(9)$ twice.", "MX" ],
[ "***", "p. 69", "$L_4(3)$", "In second lifting order row for double cover, insert $6$ in $6Q$ column and delete entry in $8J$ column; $\\chi_{41}$ is $3$ on $6Q$ and $0$ on $8J$.", "CH" ],
[ "***", "p. 69", "$L_4(3)$", "On $40A$--$D$, swap $\\chi_{37}$ with $\\chi_{38}$ and $\\chi_{39}$ with $\\chi_{40}$.", "CH" ],
[ "***", "p. 69", "$L_4(3)$", "Swap $\\chi_{43}, \\chi_{44}$ on $2.G.2_2$.", "CH" ],
[ "NEW", "p. 69", "$L_4(3)$", "Abstract Specification of 5th maximal subgroup is $N(3A_{16}B_{12}C_{12})$, and of 8th is $N(2A_6B_9)$." ],
[ "***", "p. 70", "$L_5(2)$", "$3'$ parts of $24A$ and $24B$ are $8B$ and $8C$ respectively.", "P" ],
[ "NEW", "p. 72", "$U_5(2)$", "Presentation for $G\\times 2$ $\\langle \\begin{picture}(80,40)(-10,10) \\put(0,10){\\line(1,0){60}} \\put(0,40){\\line(1,0){30}} \\put(0,10){\\line(0,1){30}} \\put(30,10){\\line(0,1){30}} \\put(0,10){\\line(1,1){30}} \\put(0,10){\\circle*{4}} \\put(30,10){\\circle*{4}} \\put(60,10){\\circle*{4}} \\put(30,40){\\circle*{4}} \\put(0,40){\\circle*{4}} \\put(-2,45){$a$}\\put(28,45){$b$}\\put(-10,8){$c$}\\put(35,15){$d$} \\put(65,8){$e$}\\end{picture} \\mid (abac)^3=(bcbd)^3=(bcacbd)^3=1\\rangle$." ],
[ "***", "p. 74", "$L_3(8)$", "$\\Gamma L_3(8) \\cong (7 \\times G).3$; $P\\Gamma L_3(8) \\cong \\Sigma L_3(8) \\cong P\\Sigma L_3(8) \\cong G.3$.", "G" ],
[ "***", "p. 74", "$L_3(8)$", "$21HI$ should be $21GH$, and have cube $7H$ and $3'$ part $7G$.", "P" ],
[ "***", "p. 74", "$^2F_4(2)'$", "Insert novelty $3^{1+2}{:}8{:}2$ in line containing 2nd maximal subgroup.", "MX" ],
[ "NEW", "p. 74", "$^2F_4(2)'$", "In Rudvalis specification, insert `base' in 5th maximal subgroup and `pentad' in 8th." ],
[ "***", "p. 77", "$Sz(32)$", "$2'$ parts of $20A$ and $20B$ are $5B$.", "P" ],
[ "***", "p. 78", "$L_3(9)$", "Change sign of $r2$ on $16EF$.", "CH" ],
[ "***", "p. 79", "$U_3(9)$", "Change sign of $r2$ on $16EF$.", "CH" ],
[ "NEW", "p. 80", "$HS$", "New presentations: for $2.G$, replace relation $(de)^2=1$ in presentation given above for $2M_{12}$ by $(ae^{dcb})^4=(de)^3=1$; and for $G$, $\\langle S_8,a\\mid a^2= [(12345)(78),a]=(14)(23)(a(56))^2=(a(67))^5=(a(678))^5=1\\rangle$." ],
[ "NEW", "p. 80", "$HS$", "Add additional remark that $HS$ is contained in $E_7(5)$." ],
[ "NEW", "p. 80", "$HS$", "In 2nd and 3rd maximal subgroups, insert `bisection' in graph specification." ],
[ "***", "p. 80", "$HS$", "In 6th maximal subgroup, replace $2244$ by $2334$.", "MX" ],
[ "NEW", "p. 80", "$HS$", "7th and 11th maximal subgroups both fix $233$-$2334$ points." ],
[ "***", "p. 80", "$HS$", "In 10th maximal subgroup, $S_5$ is non-split in $G$ (though split in $G.2$).", "MX" ],
[ "NEW", "p. 82", "$J_3$", "Insert two remarks: $J_3$ is contained in $E_6(4)$ and $E_8(2)$, and $\\chi_9$ has Schur index $2$ over $G$." ],
[ "***", "p. 82", "$J_3$", "In 7th maximal subgroup, replace $3\\times 3^2$ by $3^{1+2}$ twice.", "MX" ],
[ "***", "p. 84", "$U_3(11)$", "On $G.2$, swap first character of degree $111$ with first character of degree $1221$.", "CH" ],
[ "***", "p. 84", "$U_3(11)$", "In $12CD$ column, replace $-2r3-1$ by $2r3-1$.", "CH" ],
[ "***", "p. 85", "$O_8^+(2)$", "In Cayley section, delete all inverses from definition of duality auto\\-mor\\-ph\\-ism.", "G" ],
[ "***", "p. 87", "$O_8^+(2)$", "$18A$ cubes to $6P$, $30A$ has fifth power and $5'$ part $6O$, and $12N$ squares to $6U$.", "P" ],
[ "***", "p. 88", "$O^-_8(2)$", "Centralizer order of $2A$ should be $184320$.", "P" ],
[ "***", "p. 88", "$O^-_8(2)$", "$12DE$ square to $6C$.", "P" ],
[ "***", "p. 89", "$^3D_4(2)$", "Insert `further' before last `by' in line after starred line.", "G" ],
[ "NEW", "p. 89", "$^3D_4(2)$", "New presentation is<<nl>>$\\langle \\begin{picture}(110,15)(-10,10) \\put(0,10){\\line(1,0){90}} \\put(0,10){\\circle*{4}} \\put(30,10){\\circle*{4}} \\put(60,10){\\circle*{4}} \\put(90,10){\\circle*{4}}\\put(-2,15){$a$}\\put(28,15){$b$}\\put(58,15){$c$}\\put(88,15){$d$} \\put(13,15){6} \\end{picture} , f \\mid f^2a=(df)^4=((ab)^2c)^6=1, f=(d(ab)^3c)^7\\rangle$." ],
[ "***", "p. 91", "$A_{12}$", "In 7th and 8th maximal subgroups replace $462a$ by $462b$.", "MX" ],
[ "***", "p. 94", "$M_{24}$", "In Hexacode section, interchange $w,\\overline{w}$ in diagram (c).", "G" ],
[ "***", "p. 96", "$M_{24}$", "In Abstract Specification of 9th maximal subgroup, replace $3A$ by $3B$.", "MX" ],
[ "NEW", "p. 97", "$G_2(4)$", "In second presentation, relation $(abf)^5=1$ is redundant." ],
[ "***", "p. 98", "$G_2(4)$", "$10CD$ have $p'$ parts $AB,BB$ respectively.", "P" ],
[ "***", "p. 98", "$G_2(4)$", "Value of $\\chi_{12}$ on $8B$ is $-1$, not $1$.", "CH" ],
[ "***", "p. 99", "$G_2(4)$", "Change sign of $i3$ in $\\chi_{4}$ and $\\chi_{5}$ on $24B$ and $24C$.", "CH" ],
[ "NEW", "p. 100", "$McL$", "5th maximal subgroup has permutation character $1a+252a+4500a+5103a+5544a$; also insert `clique' in graph specification." ],
[ "NEW", "p. 100", "$McL$", "6th maximal subgroup also fixes $223$-$233$3-point." ],
[ "***", "p. 100", "$McL$", "In 7th maximal subgroup, replace $2333$ by $2344$.", "MX" ],
[ "NEW", "p. 100", "$McL$", "In 8th maximal subgroup, insert `$35$-point subgraph' in graph specification." ],
[ "NEW", "p. 100", "$McL$", "In 9th and 10th maximal subgroups, insert `$7$-point coclique' in graph specification." ],
[ "***", "p. 101", "$McL$", "In $3.G.2$, `fus ind' entries for $\\chi_{31}, \\chi_{32}, \\chi_{43}, \\chi_{44}$ should all read `$*\\ +$'.", "FI" ],
[ "***", "p. 104", "$A_{13}$", "In 6th maximal subgroup, replace $429bb$ by $429bc$.", "MX" ],
[ "***", "p. 104", "$He$", "In Monster section, line 3, replace $G$ by $G.2$.", "G" ],
[ "***", "p. 104", "$He$", "In 2nd maximal subgroup, replace $6272a$ by $1920a+4352a$.", "MX" ],
[ "M", "p. 105", "$He$", "Move end of $\\chi_2$ to right." ],
[ "***", "p. 107", "$O_7(3)$", "Change sign of $r2$ in $\\chi_{69}$ and $\\chi_{87}$ on $8C$ and $24A$.", "CH" ],
[ "***", "p. 110", "$S_6(3)$", "Change $p'$ part of $6M$ from $BC$ to $CB$.", "P" ],
[ "***", "p. 111", "$S_6(3)$", "Change class names $12N$--$R$ in $G.2$ to $12O$--$S$. This also applies on page 113.", "P" ],
[ "***", "p. 113", "$S_6(3)$", "Replace $18E$ by $18EF$.", "P" ],
[ "***", "p. 115", "$U_6(2)$", "7th maximal subgroup is $2^{4+8}{:}(S_3\\times A_5)$.", "MX" ],
[ "NEW", "p. 115", "$U_6(2)$", "7th maximal subgroups has Abstract Specification $N(2A_5B_{10})$." ],
[ "NEW", "p. 115", "$U_6(2)$", "15th maximal subgroup is $3^{1+4}{:}(Q_8\\times Q_8){:}S_3$." ],
[ "***", "p. 115", "$U_6(2)$", "In 15th maximal subgroup, delete Leech Specification.", "MX" ],
[ "***", "p. 117", "$U_6(2)$", "Swap $\\chi_{68}$ with $\\chi_{75}$ in $G.2$.", "CH" ],
[ "***", "p. 118", "$U_6(2)$", "Replace $1$ by $10$ in $6.G$ lifting orders of $10A$.", "P" ],
[ "***", "p. 120", "$U_6(2)$", "Indicator of $\\chi_{31,32,33}$ in $G.3$ is $+$, not $+\\!\\circ\\!\\circ$.", "FI" ],
[ "***", "p. 122", "$R(27)$", "$26ABDE$ square to $13EFBC$ respectively.", "P" ],
[ "***", "p. 122", "$R(27)$", "Swap character values on $9E,9F$; also on $18A,18B$.", "CH" ],
[ "NEW", "p. 123", "$R(27)$", "The product law may be changed so that $(x,y,z)(X,Y,Z)=(x+X,y+Y+X.x^s,z+Z+yX-xY-X.x^{s+1})$ with the normalizing element $k$ acting as shown. With the Sylow 3-subgroup $S$ acting on itself by left multiplication, the full group can then be generated by adjoining an element acting on $S\\cup\\{\\infty\\}$ (where $\\infty$ is fixed by $\\langle S,k\\rangle)$ that swaps $\\infty$ with $(0,0,0)$ and other points $(a,b,c)$ with $(uw^{-1},vw^{-1},cw^{-1})$ where $u=(ab)^s-c^s+a.b^2+bc-a^{2s+3}$; $v=b.a^2-ac+b^s-a^{s+3}$; $w=abc-au+b.a^{s+3}-(ab)^2-b^{s+1}-c^2$." ],
[ "NEW", "p. 123", "$S_8(2)$", "4th maximal subgroup has permutation character $1a+51a+135a+918a+1190a$ and Abstract Specification $N(2^6)=N(2B_{35}D_{28})$." ],
[ "NEW", "p. 126", "$Ru$", "Insert remark that $Ru$ is contained in $E_7(5)$." ],
[ "NEW", "p. 126", "$Ru$", "In 10th maximal subgroup, insert `pentad' in graph specification." ],
[ "***", "p. 128", "$Suz$", "Change sign of $i3$ in $\\chi_{7}$, $\\chi_8$, $\\chi_{18}$, $\\chi_{19}$, $\\chi_{21}$, $\\chi_{22}$ on $6B$, $6C$.", "CH" ],
[ "M", "p. 129", "$Suz$", "Insert $\\chi_{32}$ and $\\chi_{39}$." ],
[ "M", "p. 129", "$Suz$", "In map, box round $3.G.2$ should be open." ],
[ "***", "p. 131", "$Suz$", "In $3.G.2$, `fus ind' entries for $\\chi_{83}$, $\\chi_{84}$, $\\chi_{129}$, $\\chi_{130}$ should all read `$*8\\ \\circ$', while those for $\\chi_{121}$, $\\chi_{122}$ should read `$*5\\ \\circ$'.", "FI" ],
[ "NEW", "p. 131", "$Suz$", "Insert remark that $\\chi_{40}$ has Schur index $2$ over $G$, as does $\\chi_{54,55}$ over the group of type $2.G.2$ shown." ],
[ "***", "p. 131", "$Suz$", "In 2nd maximal subgroup, replace $2_3$ by $2'_3$.", "MX" ],
[ "***", "p. 132", "$O$'$N$", "In `Graph' section replace $52316$ by $52136$.", "G" ],
[ "NEW", "p. 132", "$O$'$N$", "In presentation for $3.G.2$, insert `involution' before `$g$'. The presentation is then genuine, and the relation $(def)^5=1$ is redundant. Another presentation for $G$ is obtained from that given above for $L_3(7){:}2$ by adjoining a generator $g$ and relations $[(cg)^2,d]=[c,(dg)^2]=afg^2=(g^bg^c)^2b=(g^eg^d)^2e=(bcdg)^5=1$." ],
[ "NEW", "p. 132", "$O$'$N$", "Insert two remarks. $\\chi_{5,6}$ has Schur index $2$ over $G.2$, as does the fusion of $\\chi_{44}$ and its conjugate over $3.G.2$; also $3.G$ has a $153$-dimensional representation over $GF(4)$, and $G$ has a 154-dimensional representation over $GF(3)$.  Explicit matrices have been constructed." ],
[ "***", "p. 134", "$Co_3$", "In 7th maximal subgroup, delete Leech specification.", "MX" ],
[ "NEW", "p. 134", "$Co_3$", "The 8th maximal subgroup is the stabilizer of a $6$-point clique in the 2-graph." ],
[ "M", "p. 136", "$O_8^+(3)$", "Insert $\\chi_4$." ],
[ "***", "p. 138", "$O_8^+(3)$", "Replace $ABC$ by $DEF$ as $2'$ parts of $18DEF$.", "P" ],
[ "***", "p. 138", "$O_8^+(3)$", "Classes $15D$--$G$ should be $15C$--$F$.", "P" ],
[ "***", "p. 140", "$O_8^+(3)$", "Replace $3^6{:}(L_4(3)\\times 2)$ by $3^6{:}(L_4(3){:}2_1)$ in $G.2_1$ column for 8th and 9th maximal subgroups.", "MX" ],
[ "***", "p. 140", "$O_8^+(3)$", "The novelties $2^7 A_8$ in $O_8^+(3):2_1$ (after the fusion markers for the 10/11th and 12/13th maximal subgroups) should be $2^6:A_8:2 = 2^6.S_8$ (the latter extension being non-split).", "MX" ],
[ "***", "p. 140", "$O_8^+(3)$", "In 19th maximal subgroup (but not 18th), change $G.2_2$ entry from $H\\times 2$ to $H{:}2$.", "MX" ],
[ "***", "p. 140", "$O_8^+(3)$", "Last maximal subgroup is $2^{1+8}.3^4.2^3$ or $2(A_4 wr 2^2).2$.", "MX" ],
[ "NEW", "p. 141", "$O^-_8(3)$", "In 1st maximal subgroup, replace first four top composition factors by $2_1,(2^2)_{122},(2^2)_{133},4$ respectively." ],
[ "***", "p. 141", "$O^-_8(3)$", "In 2nd maximal subgroup, replace $2U_4(3).D_8\\times 2$ by $2(U_4(3).D_8\\times 2)$.", "MX" ],
[ "***", "p. 141", "$O^-_8(3)$", "In 4th maximal subgroup, replace $3^{3+6}{:}(L_3(3)\\times 2^2)$ by $3^{3+6}{:}(L_3(3)\\times 4)$.", "MX" ],
[ "NEW", "p. 141", "$O^-_8(3)$", "In 9th maximal subgroup, replace $L_2(81){:}2^2$ and $L_2(81){:}(2\\times 4)$ by $2\\times L_2(81){:}2$ and $2\\times L_2(81){:}4$; also Abstract specification is $C(2H)$." ],
[ "***", "p. 144", "$O^+_{10}(2)$", "$8M$ squares to $4I$, not $4H$.", "P" ],
[ "***", "p. 147", "$O^-_{10}(2)$", "In 7th maximal subgroup, insert novelty $M_{12}{:}2$ in $G.2$.", "P" ],
[ "***", "p. 147", "$O^-_{10}(2)$", "Insert new maximal subgroup $3^5{:}2^4{:}S_5$ at bottom of table, extending to $S_3 wr S_5$ in $G.2$. This has order $466560$, index $53616640$, and specifications $N(3^5)=N(3A_5BC_{16}D_{20}E_{40}F_{40})$ and $O^-_2(2) wr S_5$.", "MX" ],
[ "***", "p. 150", "$O^-_{10}(2)$", "Insert $1$ as value of $\\chi_{27}$ on $9A$.", "CH" ],
[ "***", "p. 152", "$O^-_{10}(2)$", "$8G$ squares to $4B$, not $4A$.", "P" ],
[ "***", "p. 160", "$Fi_{22}$", "In this and the following 3 pages, 5 columns between $10C$ and $12O$ have been omitted from the character table. For details of omissions see p.~\\pageref{F22}.", "CH" ],
[ "***", "p. 160", "$Fi_{22}$", "Replace $24F$ by $F*$.", "P" ],
[ "M", "p. 160", "$Fi_{22}$", "Insert $\\chi_3$." ],
[ "NEW", "p. 163", "$Fi_{22}$", "Maximal subgroups are complete. Fifth maximal subgroup is $N(2^{10}) = N(2A_{22}B_{231}C_{770})$." ],
[ "***", "p. 163", "$Fi_{22}$", "In permutation character for 5th maximal subgroup, replace $320320a$ and $750750a$ by $32032a$ and $75075a$.", "MX" ],
[ "***", "p. 163", "$Fi_{22}$", "In map replace $48$ by $53$.", "MP" ],
[ "NEW", "p. 166", "$HN$", "Presentation for $G.2$ is<<nl>>$\\langle S_{12},a\\mid a^2=[a,(01234)]=[a,(EX987)]=[a,(45)](03)(12)=[a,(67)](8E)(9X)=(a(56))^5=1\\rangle$." ],
[ "NEW", "p. 166", "$HN$", "Insert remark that $\\chi_{42}$ has Schur index $2$ over $G$." ],
[ "NEW", "p. 166", "$HN$", "In 13th maximal subgroup, insert `$K_{3,3}$' in graph specification." ],
[ "M", "p. 168", "$F_4(2)$", "Insert $\\chi_1$." ],
[ "***", "p. 169", "$F_4(2)$", "$40B$ fifth powers to $8M$, not $8L$.", "P" ],
[ "***", "p. 170", "$F_4(2)$", "In Real section, change last sentence of second paragraph to the following: Then the special vectors are the $S_8(2)$-images of those with $x_U=2,x_*=x_V=0$ or $x_*=x_U=x_V=1$, where $U$ ranges over sets of type $1_0,5_4,0_5,4_1$; and $V$, given cyclic orderings of $A$ and $A'$, ranges over those of type $3_2$ which intersect just one of $A,A'$ in a consecutive block, and those of type $2_3$ which do not do so. Note that half the possible cyclic orderings make a vector of the second type special, but one of the first type is special for any ordering. The $S_8(2)$-stabilizers of the two vectors are respectively $S_4(2) wr 2$ ($=S_6 wr 2$) and $S_4(4){:}2$.", "G" ],
[ "NEW", "p. 170", "$F_4(2)$", "Maximal subgroups are complete." ],
[ "***", "p. 170", "$F_4(2)$", "Index of $L_3(2) wr 2$ is 58657996800.", "MX" ],
[ "M", "p. 170", "$F_4(2)$", "Interchange 13th and 14th maximal subgroups." ],
[ "***", "p. 170", "$F_4(2)$", "Replace 34 by 29 in map.", "MP" ],
[ "M", "p. 171", "$F_4(2)$", "Move first two columns of page 172 to page 171, to match pages 167--8." ],
[ "M", "p. 172", "$F_4(2)$", "Insert $\\chi_{99}$." ],
[ "***", "p. 174", "$Ly$", "Order of $2\\udot A_{11}$ is $39916800$.", "MX" ],
[ "NEW", "p. 177", "$Th$", "Delete `Status' column and query before 12th maximal subgroup, and replace `Some' by `Maximal'.<<nl>>$S_5,L_2(19){:}2$ are maximal subgroups, and the list is complete after inserting $L_3(3)$, with order $5616$, index $16158465792000$ and Abstract Specification $N(2A,3B,3B,4B,6C,8B,13A)$." ],
[ "NEW", "p. 177", "$Fi_{23}$", "Maximal subgroups are complete after modifications shown below." ],
[ "***", "p. 177", "$Fi_{23}$", "Delete last maximal subgroup.", "MX" ],
[ "NEW", "p. 177", "$Fi_{23}$", "Add new maximal subgroup $L_2(23)$, with order $6072$ and index $673496454758$." ],
[ "***", "p. 178", "$Fi_{23}$", "Replace $+$ by $\\circ$ in indicator of $\\chi_{29}$.", "FI" ],
[ "***", "p. 179", "$Fi_{23}$", "Change power maps/$p'$ parts on $12G$, $12L$, $16A$, $18A$--$G$, $20B$, $36A$ to $IB/AB$, $LD/CD$, $C/A$, $CB/CA$, $BB/BA$, $CH/CC$, $AH/AC$, $BD/BB$, $CD/CB$, $DH/DC$, $BC/AC$, $DE/AB$ respectively.", "P" ],
[ "***", "p. 180", "$Co_1$", "In Orthogonal (2) section, replace $2^{12}$ by $2^{11}$ in stabilizer of third type of $444$.", "G" ],
[ "***", "p. 181", "$Co_1$", "The second type 11 vector has shape $u_3+2v_4 (w_3)$.", "G" ],
[ "***", "p. 181", "$Co_1$", "In 4th line from bottom, $i^1=i$.", "G" ],
[ "***", "p. 182", "$Co_1$", "Middle of page, replace $D_{2n+1}$ by $D_{2n}$.", "G" ],
[ "NEW", "p. 183", "$Co_1$", "Presentations shown are proved. Another for $2.Co_1$: adjoin to that for $Co_2$ a new node labelled $h$, joined only to $a$ (unlabelled), $b$ (label $4$) and $f$ (label $6$), and add relations $(adfh)^3=d(bh)^2=d(aeh)^3=1$; for $Co_1$ adjoin also $(adefcefgh)^{39}=1$." ],
[ "***", "p. 183", "$Co_1$", "Index of 3rd maximal subgroup is $8292375$.", "MX" ],
[ "***", "p. 183", "$Co_1$", "15th maximal subgroup, not 14th, corresponds to code in $(3_+^{1+2}){:}4$.", "MX" ],
[ "***", "p. 183", "$Co_1$", "Delete 18th and 20th maximal subgroups (the two $N(3C^2)$'s).", "MX" ],
[ "***", "p. 183", "$Co_1$", "23rd maximal subgroup should be $5^2{:}2A_5$.", "MX" ],
[ "M", "p. 185", "$Co_1$", "Insert $\\chi_5$ and subscript in $\\chi_{56}$." ],
[ "***", "p. 186", "$Co_1$", "Interchange indicators of $\\chi_{141}$ and $\\chi_{143}$.", "FI" ],
[ "NEW", "p. 190", "$J_4$", "maximal subgroups are complete after insertions shown below." ],
[ "C", "p. 190", "$J_4$", "Transpose 2nd and 3rd maximal subgroups." ],
[ "NEW", "p. 190", "$J_4$", "Insert new maximal subgroups: $M_{22}{:}2$ after 5th, and $U_3(3)$ after 8th; both may be specified as vector stabilizers." ],
[ "NEW", "p. 191", "$^2E_6(2)$", "Replace `For a possible presentation' by `For presentations'." ],
[ "NEW", "p. 191", "$^2E_6(2)$", "Maximal subgroups are complete." ],
[ "***", "p. 191", "$^2E_6(2)$", "11th maximal subgroup is $2^{3+4+12+12}{:}(A_5\\times L_3(2))$.", "MX" ],
[ "NEW", "p. 191", "$^2E_6(2)$", "14th maximal subgroup is $U_3(8){:}3_1$." ],
[ "NEW", "p. 191", "$E_6(2)$", "Maximal subgroups are complete if $(7\\times ^3D_4(2)){:}3$, $7^3{:}3^{1+2}{:}2A_4$ and two classes of $U_3(3){:}2$ are added.<<nl>>In 3rd and 4th maximal subgroups, $[2^{25}]$ may be replaced by $2^{5+20}$. Maximal subgroups of $G{:}2$ are $H{:}2$ for all maximal subgroups $H$ of $G$ except the $3$ pairs of fusing classes; plus novelties $[2^{24}]{:}O_8^+(2){:}2$ and $O^+_{10}(2){:}2$ (from 1st and 2nd maximal subgroups) and $[2^{31}]{:}(S_3\\times S_3\\times L_3(2)){:}2$ (from 3rd and 4th maximal subgroups)." ],
[ "***", "p. 194", "$^2E_6(2)$", "$8C1$ squares to $4O$, not $4I$.", "P" ],
[ "M", "p. 195", "$^2E_6(2)$", "Insert $\\chi_{99}$ twice." ],
[ "M", "p. 196", "$^2E_6(2)$", "Insert $\\chi_{192}$ and $\\chi_{193}$." ],
[ "***", "p. 201", "$Fi'_{24}$", "$18D$ cubes to $6C$.", "P" ],
[ "***", "p. 203", "$Fi'_{24}$", "The $13$th power and $13'$ part of $78A$ and $78B$ are $6L$.", "P" ],
[ "M", "p. 206", "$Fi'_{24}$", "Delete $\\chi_{109}$ on right." ],
[ "***", "p. 206", "$Fi'_{24}$", "In $3.G.2$, `fus ind' entries for $\\chi_{120,121,140,141}$ should all read `$*8\\ \\circ$'; those for $\\chi_{128,129}$ should read `$*5\\ \\circ$'; and those for $\\chi_{166,167}$ should read `$.\\ +\\!2$' and `$*{}*$' respectively with a line joining `$.$' to `$*{}*$'.", "FI" ],
[ "M", "p. 207", "$Fi'_{24}$", "Delete $\\chi_{109}$ twice." ],
[ "***", "p. 207", "$Fi'_{24}$", "In Modulo $3$ section, delete from after the comma on the second line, and replace by `and an algebra on which $I$ is the identity, obtained by modifying the above algebra, for which the only non-zero products of basis vectors are $e_i * e_i=e_i$ $e_i * e_A=e_A * e_i=-e_A$ ($i\\in [A]$) $e_A * e_A=-e_A * e_{-A}=\\sum_{i\\in [A]}e_i$ $e_A * e_B=e_{AB}$ ($[AB]$ an octad).", "G" ],
[ "NEW", "p. 207", "$Fi'_{24}$", "In the notation of page 232, $\\langle Y_{442}\\mid S\\rangle$ is a genuine presentation for $3.G.2$." ],
[ "NEW", "p. 207", "$Fi'_{24}$", "Maximal subgroups are complete if two new pairs, both interchanged by the outer automorphism, are added. These are $U_3(3){:}2$ (with specification $N(2B,3D,3E,4C$, $4C,6J,7B,8C,12M))$ and $L_2(13){:}2$ (with specification $N(2B,3D,6J,7B,13A))$." ],
[ "***", "p. 210", "$B$", "$2'$ parts of $24H$ and $24L$ are $3B$.", "P" ],
[ "NEW", "p. 216", "$B$", "In the notation of p. 232, $\\langle Y_{433}\\mid S\\rangle$ is a genuine presentation for $2\\times 2.G$." ],
[ "M", "p. 216", "$B$", "In $2.G$, negate all character values on $6G,10D,12F,20G,24L,36C$." ],
[ "***", "p. 217", "$B$", "$N(3^6)$ should be $3^6{:}(2\\times L_4(3))\\udot 2^2$.", "MX" ],
[ "***", "p. 217", "$B$", "Supergroup of $N(23AB)$ is non-split, as shown in $N(2B)$.", "MX" ],
[ "NEW", "p. 217", "$B$", "$L_2(31)$ is another supergroup of $N(31AB)$." ],
[ "NEW", "p. 217", "$B$", "New maximal subgroups $(S_6\\times L_3(4){:}2){:}2$, $(S_6\\times S_6){:}4$, $L_2(31)$ and $L_2(49)\\udot 2_3$." ],
[ "M", "p. 221", "$M$", "Insert $-2$ as value of $\\chi_{81}$ on $23A$." ],
[ "M", "p. 226", "$M$", "Insert $0$ as value of $\\chi_{109}$ on $41A$." ],
[ "***", "p. 231", "$M$", "Replace $F_{5|2-}$ by $F_{4|2-}$ in middle of page.", "G" ],
[ "NEW", "p. 231", "$M$", "It should be noted that replication formulae other than the duplication formula (as stated) are needed to derive expressions for all $H_i$ in terms of the first few." ],
[ "NEW", "p. 231", "$M$", "$L_2(41)$ and $J_1$ are not involved, but $L_2(19)$, $L_2(31)$, $L_2(49)$ are (in $Th$, $B$, $B$ respectively).<<nl>>NO, $L_2(41)$ IS involved, see the 2nd part of the improvements." ],
[ "NEW", "p. 232", "$M$", "All conjectures are now proved." ],
[ "NEW", "p. 233", "$M$", "Delete all queries; but in line beginning $Q_{222}$ the relations are not sufficient to present the group named." ],
[ "NEW", "p. 234", "$M$", "Permutation character on transpositions is $\\chi_{1}+\\chi_2+\\chi_4+\\chi_5+\\chi_9+\\chi_{14}+\\chi_{21}+\\chi_{34}+\\chi_{35}$." ],
[ "***", "p. 234", "$M$", "$N(5B^4)$ is $5^4{:}(3\\times 2\\udot L_2(25)){:}2_2$ with outer involution inverting $3$.", "MX" ],
[ "***", "p. 234", "$M$", "New maximal $7$-local subgroup $N(7B^2)=7^2{:}SL_2(7)$.", "MX" ],
[ "NEW", "p. 234", "$M$", "$41{:}40$ is maximal; delete $L_2(41){:}2$ as possible supergroup." ],
[ "***", "p. 235", "$E_8(2)$", "Replace $3\\udot U_9(2){:}S_3$ by $(3\\times U_9(2)){:}2$.", "MX" ]
   ]
  },
  {
   "source": "1",
   "section": "Additional information",
   "entries": [
[ "NEW", "p. 238", "", "Delete lines 5--7 and replace $?$ by $-$ in bottom line, as $J_1$ is not involved in $M$." ],
[ "NEW", "p. 238", "", "Containments and involvements of sporadic groups in exceptional groups are now known. Here are the minimal containments for each sporadic group that has any; in the starred cases the relevant subgroup must be maximal. (It is not asserted that there are no other containments as maximal subgroups.) $M_{11}$: In $F_4(11)$, $E_6(p)$ or $^2E_6(p)$ according as $p^5 \\equiv 1$ or $-1$ (mod $11$). $M_{12}$: In $^2E_6(2)$, $E_6(5)$*, $E_7(p)$ for $(p,10) = 1$. $J_1$: In $G_2(11)$*. $M_{22}$: In $^2E_6(2)$, $E_7(5)$. $J_2$: In $G_2(4)$, $E_7(p)$ for odd $p$. $HS$: In $E_7(5)$*. $J_3$: In $E_6(4)$*, $E_8(2)$. $Ru$: In $E_7(5)$*. $Fi_{22}$: In $^2E_6(2)$*. $Th$: In $E_8(3)$*. It is also known that $L_2(37)$ and $L_2(61)$ are subgroups of $E_7(\\CC)$ and $E_8(\\CC)$ respectively, hence of $E_7(q)$ and $E_8(q)$ for suitable $q$ (in any characteristic)." ],
[ "***", "p. 239", "", "$L_2(8)$ has multiplier $1$, not $2$.", "I" ],
[ "***", "p. 240", "", "In $U_4(5)$ row, replace $2^5$ and $5^4$ by $2^7$ and $5^6$.", "I" ],
[ "***", "p. 241", "", "In $L_3(29)$ and $G_2(7)$ rows, replace $871$ by $13.67$ and $817$ by $19.43$.", "I" ]
   ]
  },
  {
   "source": "1",
   "section": "Bibliography",
   "entries": [
[ "NEW", "p. 243", "", "In lines 7--8 of list, replace 2nd by 4th and $1965$ by $1984$." ],
[ "M", "p. 244, line 5", "$M_{12}$", "tetracode --- MOG." ],
[ "M", "p. 244, line 6", "$M_{12}$", "exceptional." ],
[ "M", "p. 244, line 8", "$M_{12}$", "$95040$." ],
[ "M", "p. 244, line 19", "$M_{12}$", "representations." ],
[ "M", "p. 244, line 20", "$M_{12}$", "90 3, (1981)." ],
[ "M", "p. 244, line 21", "$M_{12}$", "vertices of the." ],
[ "M", "p. 244, line 3", "$J_1$", "215--247." ],
[ "M", "p. 245, line 16", "$J_1$", "1978." ],
[ "M", "p. 245, line 17", "$J_1$", "1981." ],
[ "M", "p. 245, line 19", "$J_1$", "(3) 19 (1969)." ],
[ "***", "p. 245, line 22", "$J_1$", "$PG(6,11)$, Proc.\\ Cambridge.", "I" ],
[ "NEW", "p. 245, line 18", "$J_2$", "Geom.\\ Dedic.\\ 20 (1986) 157--173." ],
[ "M", "p. 246, line 4", "$HS$", "group." ],
[ "M", "p. 246, line 5", "$HS$", "the." ],
[ "M", "p. 246, line 6", "$HS$", "$44,352,000$." ],
[ "M", "p. 246, line 16", "$HS$", "Replace `One' by `On the' and `ed.'\\ by `eds.'." ],
[ "NEW", "p. 246, line 22", "$HS$", "J. London Math.\\ Soc.\\ (2) 32 (1985) 460--466." ],
[ "M", "p. 246, line 2", "$J_3$", "A. Rudvalis." ],
[ "M", "p. 246, line 3", "$J_3$", "Algebra." ],
[ "NEW", "p. 247, line 10", "$He$", "add `Canberra, 1973, ed.\\ M. F. Newman, 448--452, in Springer Lecture Notes no. 372'." ],
[ "NEW", "p. 247, line 12", "$He$", "J. London Math.\\ Soc.\\ (2) 32 (1985) 460--466." ],
[ "M", "p. 248, line 9", "$Suz$", "Delete `finite' from title of paper." ],
[ "NEW", "p. 248, line 7", "$O'N$", "J. Algebra\\ 108 no. 2 (1987) 310--316." ],
[ "NEW", "p. 248, line 9", "$O'N$", "J. Algebra\\ 97 (1985) 467--473." ],
[ "NEW", "p. 248, line 10", "$O'N$", "J. Fac.\\ Sci.\\ Univ.\\ Tokyo, Sec.\\ 1A 32 No.\\ 1 (1985) 105--141." ],
[ "M", "p. 248, line 5", "$Co_3$", "Worboys." ],
[ "M", "p. 249, line 3", "$HN$", "eds.." ],
[ "NEW", "p. 249, line 7", "$HN$", "J. Algebra\\ 103 (1986) 362--376." ],
[ "NEW", "p. 249, line 2", "$Ly$", "Math.\\ Ann.\\ 267 (1984), 519--535." ],
[ "NEW", "p. 249, line 3", "$Ly$", "Math.\\ Ann.\\ 272 (1985), 29--39." ],
[ "NEW", "p. 249, line 7", "$Ly$", "(3) 97 (1985) 433--436." ],
[ "M", "p. 249, line 9", "$Fi_{23}$", "by a; 44 (1977)." ],
[ "***", "p. 250, line 3", "$Co_1$", "$8,315,553,613,086,720,000$.", "I" ],
[ "M", "p. 250, line 20", "$Co_1$", "Replace 1--7 by 523--524." ],
[ "***", "p. 250, line 3", "$J_4$", "$86,775,571,046,077,562,880$.", "I" ],
[ "NEW", "p. 250, line 4", "$Fi'_{24}$", "Paper has title `On the group $Fi_{24}$' and is in Geom.\\ Dedic.\\ 25 (1988) (Geometries and Groups issue) 483--501." ],
[ "M", "p. 251, line 1", "$M$", "Intelligencer." ],
[ "NEW", "p. 251, line 2", "$M$", "Replace `simplified' by `simple'. Reference is 79 (1985) 513--540." ],
[ "NEW", "p. 251, line 19", "$M$", "Replace `Proceedings...' by `Finite Groups -- Coming of age, ed.\\ J. McKay, 1985 (Contemporary Mathematics Series, Vol 45), pp 271--285'." ],
[ "NEW", "p. 251, line 26", "$M$", "78 (1984) 491--499." ]
   ]
  },
  {
   "source": "2",
   "section": "Introduction",
   "entries": [
[ "C", "Page iv", "", "Top, insert ``page'' before ``number''." ],
[ "M", "Page vii", "", "Top left, change head of Section 1 to ``Preliminaries''." ],
[ "C", "Page vii", "", "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$\\ldots$generate a multiplicative group which is$\\ldots$''." ],
[ "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''.", "I" ],
[ "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", "Page xxix", "", "In Section 18, specify notational convention where both coset and cohort are\\break ``squashed''." ],
[ "M", "Page xxxi", "", "Section 1, replace ``product'' by ``produce''." ]
   ]
  },
  {
   "source": "2",
   "section": "The Groups",
   "entries": [
[ "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 $Q_8\\times S_3$ by $(Q_8\\times 3).2$.", "MX" ],
[ "***", "Page 17", "L_2(25)", "Change element orders of $4C$, $12C$, $12D$ in $4.G.2_3$ to $4$, $12$, $12$.", "P" ],
[ "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 $x=y=0$''.", "G" ],
[ "C", "Page 29", "L_2(32)", "Singularize ``Presentations''." ],
[ "***", "Page 30", "U_3(4)", "Change class names $10A$--$B$ in $G.2$ to $10E$--$F$.", "P" ],
[ "NEW", "Page 30", "U_3(4)", "In 3rd Max, $N(5^2)$ is also $N(5A$--$D_3EF_3)$." ],
[ "C", "Page 31", "M_{12}", "In {\\bf MINIMOG} section close bracket in 6th line." ],
[ "NEW", "Page 33", "M_{12}", "In 9th Max, replace $H.2$ by $M_8.(S_4\\times 2)$." ],
[ "M", "Page 36", "J_1", "Raise most of bottom line in ``Graph'' section." ],
[ "C", "Page 36", "J_1", "Replace $=$ by $\\cong$ in presentation." ],
[ "C", "Page 36", "J_1", "In 6th Max, replace $D_6$ by $S_3$." ],
[ "NEW", "Page 42", "J_2", "In 4th Max, replace $H.2$ by $2^{2+4}.(S_3\\times S_3)$." ],
[ "***", "Page 52", "U_4(3)", "In 6th Max, replace $3^{1+4}_+.(2S_4 \\times 2)$ by $3^{1+4}_+.4S4$.", "MX" ],
[ "NEW", "Page 53", "U_4(3)", "1st Max is $N(3^4)=N(3A_{10}B_{15}C_{15})$." ],
[ "***", "Page 55", "U_4(3)", "Change class name $12J$ in $G.2_3$ to $10K$. This also applies on page 57 and 59.", "P" ],
[ "***", "Page 60", "G_2(3)", "In ``$14$-dimensional'' construction, in 5th line image of $e_i$ should be divided by $3$.", "G" ],
[ "C", "Page 66", "U_3(8)", "Remove the note from the map that had been suggested in Appendix 2 of {\\bf 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 $\\chi_{31}$--$\\chi_{34}$ 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 $N(7A$--$F_3GH_2IJK_3)$." ],
[ "***", "Page 77", "Sz(32)", "In ``Suzuki'' construction, insert before comma in 2nd line ``and such that when $t=0$ we have $x=y=0$''.", "G" ],
[ "NEW", "Page 79", "U_3(9)", "In 4th Max, $N(5^2)$ is also $N(5A$--$D_3EF_3)$." ],
[ "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", "HS", "Replace $*7$ by $*2$ in $\\chi_{42}$." ],
[ "***", "Page 82", "J_3", "In ``Unitary'' construction, the definition of $C_Z$ only applies when $z\\ne 0$: the image of $e(0)$ is $e(0)$.", "G" ],
[ "C", "Page 83", "J_3", "Replace $r6\\ *$ by $r6\\ -\\!r6$ in $\\chi_9$." ],
[ "C", "Page 89", "O_8^-(2)", "Pluralize ``Presentation''." ],
[ "***", "Page 89", "^3D_4(2)", "In ``Jordan'' construction, replace top row (but not 1st column) occurrences of ${1\\over 4}s$ by ${1\\over 4}\\overline{s}$ in $768$ matrix orbit.", "G" ],
[ "M", "Page 89", "^3D_4(2)", "In line after display of roots in same section, delete comma before slash." ],
[ "NEW", "Page 89", "^3D_4(2)", "New presentation for $G$ is linear Coxeter graph \\hfill\\break $\\langle a6b3c3d\\mid ((ab)^2c)^6,(a^{bc}(a^{b^cb^a}a^{bcd})^2)^3,(abcd)^{21}\\rangle$.  Omitting the last relator gives a presentation for $S_4\\times G$." ],
[ "NEW", "Page 89", "^3D_4(2)", "In 7th Max, $N(7^2)$ is also $N(7A$--$C_4D_4)$." ],
[ "***", "Page 97", "G_2(4)", "In 2nd Max, delete entry in last column.", "MX" ],
[ "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''." ],
[ "***", "Page 109", "O_7(3)", "In 14th Max, replace $2E$ by $2F$.", "MX" ],
[ "C", "Page 113", "S_6(3)", "Move ``Specifications'' to right." ],
[ "NEW", "Page 113", "S_6(3)", "2nd Max is $N(3^6)=N(3AB_{13}C_{39}D_{78}E_{234})$." ],
[ "NEW", "Page 113", "S_6(3)", "3rd Max is $N(3^3)=N(3AB_4C_3D_6)$." ],
[ "***", "Page 113", "S_6(3)", "In 7th Max, extensions are split.", "MX" ],
[ "NEW", "Page 115", "U_6(2)", "$N(2^9)$ is also $N(2A_{21}B_{210}C_{280})$." ],
[ "C", "Page 115", "U_6(2)", "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 $3^{3+3+3}$.", "MX" ],
[ "C", "Page 123", "R(27)", "Insert separator between $R(27)$ and $S_8(2)$." ],
[ "C", "Page 123", "S_8(2)", "Pluralize ``a presentation''." ],
[ "NEW", "Page 123", "S_8(2)", "7th Max is $N(2^6)=N(2A_7B_7C_{21}E_{28})$." ],
[ "C", "Page 129", "Suz", "Replace $r10\\ *$ by $r10\\ -\\!r10$ in $\\chi_{40}$." ],
[ "C", "Page 131", "Suz", "Pluralize ``Presentation''." ],
[ "***", "Page 131", "Suz", "12th Max is non-split.", "MX" ],
[ "C", "Page 140", "O_8^+(3)", "Pluralize ``a presentation''." ],
[ "NEW", "Page 140", "O_8^+(3)", "Maxes 7-9 are also $N(3^6)$." ],
[ "***", "Page 140", "O_8^+(3)", "In 8th and 9th Maxes, replace $L_4(3)\\times 2$ by $L_4(3)\\!:\\!2_1$.", "MX" ],
[ "***", "Page 140", "O_8^+(3)", "The novelties $2^7 A_8$ in $O_8^+(3)\\!:\\!2_1$ (after the fusion markers for the 10/11th and 12/13th maximal subgroups) should be $2^6\\!:\\!A_8\\!:\\!2 = {2^6}^{\\bf .}S_8$ (the latter extension being non-split).", "MX" ],
[ "NEW", "Page 146", "O_{10}^+(2)", "In 2nd Max, $N(2^8)$ is also $N(2A_{135}B_{120})$." ],
[ "NEW", "Page 146", "O_{10}^+(2)", "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", "Page 147", "O^-_{10}(2)", "In 5th Max, $N(2^6)$ is also $N(2A_{35}C_{28})$." ],
[ "NEW", "Page 147", "O^-_{10}(2)", "In 6th Max, replace $N(2^3)$ by $N(2A^3)$." ],
[ "C", "Page 147", "O^-_{10}(2)", "In last Max, $N(3A_2D_2)$ 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", "Fi_{22}", "In the additional five columns, on p.~310 of {\\bf ABC}, the centralizer orders of $12L$, $12M$ and $12N$ are $3456$, $1296$ and $576$ respectively.", "P" ],
[ "***", "Page 163", "Fi_{22}", "7th Max is $(2\\times 2_+^{1+8})\\!:\\!U_4(2):2$ extending to $H\\!:\\!2$.", "MX" ],
[ "NEW", "Page 163", "Fi_{22}", "In 8th Max, replace $N(2^4)$ by $N(2B^4)$." ],
[ "C", "Page 172", "F_4(2)", "Replace $-0$ by $0$ in $\\chi_{96}$ on $10A$." ],
[ "NEW", "Page 177", "Fi_{23}", "In 6th Max, $N(2^{11})$ is also $N(2A_{23}B_{253}C_{1771})$." ],
[ "NEW", "Page 177", "Fi_{23}", "In 11th Max, $N(2^6)$ is also $N(2A_7B_{21}C_{35})$." ],
[ "C", "Page 182", "Co_1", "Near bottom, pluralize ``portion'', ``group'' and ``parenthesis''." ],
[ "***", "Page 183", "Co_1", "Maximal subgroup $5^3:(4 \\times A_5).2$ is in fact $5^3:(4 \\times S_5)$.", "MX" ],
[ "C", "Page 191", "^2E_6(2)", "Move maximal subgroup $N(3C)$ above $U_3(8).3_1$." ],
[ "***", "Page 191", "^2E_6(2)", "$N(3C)$ is $3^{1+6}\\!:\\!2^{3+6}\\!:\\!3^2\\!:\\!2$.", "MX" ],
[ "C", "Page 191", "^2E_6(2)", "In last Max, replace $3^2Q_8$ 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 $(S_6\\times L_3(4)\\!:\\!2)\\!:\\!2$ quoted in the original list of amendments, the 1st ``$:\\!2$'' is in fact ``$:\\!2_2$''." ],
[ "C", "Page 230", "M", "The symbol ${\\Bbb M}$ is now used for the Monster." ],
[ "C", "Page 230", "M", "Pluralize ``Presentation''." ],
[ "NEW", "Page 230", "M", "New presentation for $G$ has generators $s$, $t$, $v$, $x$ and relators $s^6$, $t^3$, $(st)^4$, $(s^2t)^4$, $(s^3t)^3$, $[s^2,(ts^2t)^2]$, $[v,ut^{-1}]$, $[v,u^3su^{-2}]$, $v^2$, $[v,v^t]$, $(vu)^{13}$, $[vx,ut^{-1}]$, $[vx,s^{u^3}]$, $x^3$, $(v^uvx)^2$, $(x^{-1}x^s)^2$, $(xt^{-1})^{12}$, $(u^{-6}xu^6s)^6(sux^{-1}u^{-1})^6s^{-1}$ and $((xv^{u^4}v^{u^{10}})^3u)^{13}$.  $s$ 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 $G\\times L_3(3)$, $G\\times G$, and $G\\times G\\times L_3(3)$ respectively." ],
[ "M", "Page 230", "M", "Replace $,$ by $.$ in last line before list of groups." ],
[ "NEW", "Page 231", "M", "A list of groups which may be involved or contained in the Monster has been compiled. See below.<<nl>>We divide all simple groups whose order divides that of ${\\Bbb M}$ into the following four types:<<nl>>(a) The following groups are subgroups of ${\\Bbb M}$ and their occurrences are completely classified: $C_p$ ($p$ prime dividing $|{\\Bbb M}|$), $A_n$ ($5\\le n\\le 12$), $L_2(q)$ ($q=7,8,11,17,19,23,25,29,31,41,49,59,71$), $L_4(3)$, $U_3(5)$, $U_5(2)$, $S_4(4)$, $S_n(2)$ ($n=6,8$), $O_7(3)$, $O_8^+(q)$ ($q=2,3$), $O_n^-(2)$ ($n=8,10$), $G_2(3)$, ${}^3D_4(2)$, ${}^2F_4(2)'$, $M_{12}$, $Fi_{23}$, $He$, $HN$.<<nl>>(b) The following groups are subgroups of ${\\Bbb M}$ but their occurrences have not been completely classified: $L_2(q)$ ($q=13,16$).<<nl>>(c) The following groups are known to be involved in ${\\Bbb M}$ but not contained: $L_2(81)$, $L_3(q)$ ($q=4,5$), $L_5(2)$, $U_4(3)$, $U_6(2)$, $O_8^-(3)$, $O_{10}^+(2)$, $G_2(4)$, $F_4(2)$, ${}^2E_6(2)$, $M_n$ ($n=22,23,24$), $Co_n$ ($n=1,2,3$), $Fi_{22}$, $Fi_{24}'$, $J_2$, $Suz$, $HS$, $McL$, $B$.<<nl>>(d) The following groups are not involved in ${\\Bbb M}$: $A_n$ ($13 \\le n\\le 32$), $L_2(q)$ ($q=27,32,47,64,125,169$, $1024$), $L_3(q)$ ($q=7,9,16,25$), $L_4(q)$ ($q=4,5,7,9$), $L_5(q)$ ($q=3,4$), $L_6(q)$ ($q=2,3,4$), $U_4(q)$ ($q=4,5,8$), $U_n(4)$ ($n=5,6$), $S_4(q)$ ($q=5,7,8,9$), $S_6(q)$ ($q=3,4,5$), $S_8(3)$, $S_n(2)$ ($n=10,12$), $O_7(5)$, $O_9(3)$, $O_{10}^+(3)$, $O_{12}^{\\pm}(2)$, $Sz(8)$, $Sz(32)$, $G_2(5)$, $J_n$ ($n=1,3$), $O'N$, $Ru$." ],
[ "NEW", "Page 234", "M", "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." ]
   ]
  },
  {
   "source": "2",
   "section": "Additional information",
   "entries": [
[ "C", "Page 239", "", "Pluralize ``Order''." ],
[ "NEW", "Page 239", "", "Insert sentence: groups whose full character tables are in the {\\bf ATLAS} are starred, while a bracketed star is appended to other {\\bf ATLAS} groups and to those for which all or part of the character table exists in {\\bf ATLAS} format. Also, insert relevant stars, with or without brackets, in pages 239-242." ],
[ "***", "Page 241", "", "Outer automorphism group of $G_2(9)$ is $4$.", "I" ]
   ]
  },
  {
   "source": "2",
   "section": "Bibliography",
   "entries": [
[ "M", "Page 247, line 3", "", "Replace 243 by 323." ]
   ]
  }
]
}

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