  
  [1X7 [33X[0;0YLarge Nilpotent Subgroups of Sporadic Simple Groups[133X[101X
  
  [33X[0;0YDate: June 6th, 2009[133X
  
  [33X[0;0YWe  show that any nontrivial nilpotent subgroup [22XU[122X in a sporadic simple group
  [22XG[122X  satisfies  [22X|U|  ⋅  |N_G(U)|  < |G|[122X. The proof uses the information in the
  [5XAtlas[105X  of  Finite  Groups [CCN+85] and the [5XGAP[105X system [GAP21], in particular
  its   Character   Table   Library [Bre22]  and  its  library  of  Tables  of
  Marks [NMP18]. (In [Vdo00], it is shown that in any finite nonabelian simple
  group [22XG[122X, any nilpotent subgroup [22XU[122X satisfies [22X|U|^2 < |G|[122X.)[133X
  
  
  [1X7.1 [33X[0;0YThe Result[133X[101X
  
  [33X[0;0YThe aim of this writeup is to show the following statement.[133X
  
  [33X[0;0Y[13XProposition[113X:  Let  [22XG[122X  be  a  sporadic  simple  group,  let [22XU[122X be a nontrivial
  nilpotent  subgroup  in  [22XG[122X,  and let [22XN_G(U)[122X denote the normalizer of [22XU[122X in [22XG[122X.
  Then [22X|U| ⋅ |N_G(U)| < |G|[122X holds.[133X
  
  [33X[0;0YThe  following  criteria are sufficient to prove this proposition. Note that
  we  are  interested  in  an  argument  that  uses only information about the
  character  tables  of  the  sporadic  simple  groups  and  of  their maximal
  subgroups.[133X
  
  [33X[0;0Y[13XLemma  1[113X: Let [22XG[122X be a nonabelian finite simple group, and suppose that [22XU[122X is a
  nontrivial nilpotent subgroup of [22XG[122X such that [22X|U| ⋅ |N_G(U)| ≥ |G|[122X holds. Let
  [22XΠ  = { p_1, p_2, ..., p_n }[122X be the set of prime divisors of [22X|U|[122X, and set [22Xn =
  ∏_{p ∈ Π} p[122X.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22XG[122X  contains  an element [22Xg[122X of order [22Xn[122X and a maximal subgroup [22XM[122X with the
        properties  [22Xg  ∈  Z(U)[122X  and  [22XN_G(U) ≤ M[122X. Set [22Xc:= gcd(|C_G(g)|_Π, |M|)[122X,
        where  [22X|C_G(g)|_Π[122X  denotes  the  largest  divisor  of the order of the
        centralizer  of [22Xg[122X in [22XG[122X whose prime divisors are elements of the set [22XΠ[122X.
        Then  we  have  [22X|U| ≤ c[122X and hence [22Xc ⋅ |M| ≥ |G|[122X, in particular [22X|M|^2 ≥
        |G|[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YIf [22X(g, M)[122X is as in part (a) then one of the following holds.[133X
  
        [8X(b1)[108X
              [33X[0;12Y[22XU[122X is normal in [22XM[122X, and the Fitting subgroup [22XFit(M)[122X of [22XM[122X satisfies
              [22X|Fit(M)| ⋅ |M| ≥ |G|[122X.[133X
  
        [8X(b2)[108X
              [33X[0;12Y[22XU[122X  is  not  normal in [22XM[122X, so [22XN_G(U)[122X is a proper subgroup of [22XM[122X, in
              particular [22X|G| ≤ |U| ⋅ |M|/2 ≤ c ⋅ |M| / 2[122X holds.[133X
  
  [8X(c)[108X
        [33X[0;6YLet  [22X(g,  M)[122X  be  as  in part (b2) and assume that [22XM[122X contains a normal
        subgroup  [22XK[122X  such that [22Xπ(M):= M/K[122X is an almost simple group with socle
        [22XS[122X,  i. e.,  [22Xπ(M)[122X  has  a nonabelian simple normal subgroup [22XS[122X such that
        [22XC_{π(M)}(S)[122X is trivial. Then either [22XU ≤ K[122X holds, and hence [22X|K| ⋅ |M| ≥
        |G|[122X, or we are in the following situation.[133X
  
        [33X[0;6YThe  group [22Xπ(U):= U K / K[122X is a nontrivial nilpotent normal subgroup of
        [22Xπ(N):=  N_G(U)  K / K[122X, and [22XH:= S ∩ π(N)[122X is a proper subgroup of [22XS[122X. The
        latter statement holds because otherwise [22XS ∩ π(U)[122X would be normal in [22XS[122X
        and  thus  would be trivial, which would imply that [22XS[122X would centralize
        [22Xπ(U)[122X.[133X
  
        [33X[0;6YAs  a  consequence, [22X|π(N)|[122X divides [22X|π(M)/S| ⋅ |H| = |π(M)| / [S:H][122X, in
        particular,  [22X[S:H]  ≤  |π(M)|  /  |π(N)|  =  |M|  / |N_G(U) K| ≤ |M| /
        |N_G(U)| ≤ |M| ⋅ |U| / |G| ≤ c / [G:M][122X holds.[133X
  
  [33X[0;0YWe will apply Lemma 1 as follows.[133X
  
  [33X[0;0YFrom  the  character  tables  of [22XG[122X and [22XM[122X, the value [22X|Fit(M)|[122X and the maximal
  possible  [22Xc[122X can be computed. If part (a) of the lemma applies then we verify
  that  part (b1)  does  [13Xnot[113X  apply,  and  that  either  (b2)  or (c) yields a
  contradiction.  Note  that  we  can  determine from the character table of [22XM[122X
  whether  [22XM[122X  has  a  normal subgroup [22XK[122X such that [22XM/K[122X is almost simple, and in
  this case we can compute the order of the socle [22XS[122X of [22XM/K[122X.[133X
  
  [33X[0;0YFor proving the nonexistence of the subgroup [22XH[122X in the situation of part (c),
  we  will  show  that all subgroups of [22Xπ(M)[122X of index up to [22Xd:= c ⋅ [π(M):S] /
  [G:M][122X  contain  [22XS[122X.  For  that,  we  will  compute the complete list of those
  possible  permutation characters of [22Xπ(M)[122X whose degree is at most [22Xd[122X, and then
  check that the kernels of these characters contain [22XS[122X.[133X
  
  [33X[0;0Y(Note  that these computations are cheap because the bound [22Xd[122X is small in the
  cases  that occur. There are easier criteria for proving the nonexistence of
  a  subgroup  of index at most [22Xd[122X in a simple group [22XS[122X, for example in the case
  [22X|S|  >  d!  /  2[122X or if the smallest nontrivial irreducible degree of [22XS[122X is at
  least [22Xd[122X; but these criteria do not suffice in our situation.)[133X
  
  [33X[0;0YWe illustrate the application of Lemma 1 with some examples.[133X
  
  [8X[22XJ_1[122X:[108X
        [33X[0;6YThe  first Janko group [22XJ_1[122X (see [CCN+85, p. 36]) has order [22X175560[122X, and
        the largest maximal subgroup has order [22X660[122X. The largest centralizer of
        a  nonidentity  element  in [22XJ_1[122X has order [22X120[122X, and [22X660 ⋅ 120 = 79200 <
        |J_1|[122X. Thus [22XJ_1[122X satisfies the proposition.[133X
  
  [8X[22XMM[122X:[108X
        [33X[0;6YFor  the Monster group [22XMM[122X (see [CCN+85, p. 234]), we read off from the
        list [Wil]  of  maximal subgroups that the only maximal subgroups [22XM[122X of
        [22XMM[122X  with  the  property [22X|M|^2 ≥ MM[122X have the structure [22X2.B[122X. Already for
        the   second   largest   maximal   subgroups,   with   the   structure
        [22X2^{1+24}.Co_1[122X, the order is smaller than the index in the Monster.[133X
  
        [33X[0;6YOnly elements [22Xg[122X from the classes [10X2A[110X, [10X2B[110X, and [10X3A[110X have the property that
        the product of [22X|2.B|[122X and the order of the centralizer of [22Xg[122X in [22XM[122X is not
        smaller  than  [22X|M|[122X.  So  [22XU[122X  can  be  only a [22X2[122X- or a [22X3[122X-subgroup of [22X2.B[122X.
        However,  the  [22X2[122X-part  and  the  [22X3[122X-part  of  [22X|2.B|[122X  are [22X2^42[122X and [22X3^13[122X,
        respectively,  which  are  smaller  than the index of [22X2.B[122X in [22XM[122X. Thus [22XM[122X
        satisfies the proposition.[133X
  
  [8X[22XFi_23[122X:[108X
        [33X[0;6YWe  show  that  no  counterexample  to  the proposition can arise from
        maximal  subgroups  [22XM[122X  of  the  type [22XO_8^+(3):S_3[122X in the Fischer group
        [22XFi_23[122X  (see [CCN+85,  p.  177]).  Several  element  centralizers  in [22XG[122X
        satisfy  Lemma 1 (a),  the largest value [22Xc[122X arises from elements in the
        class  [10X6B[110X, whose centralizers have order [22X2^8 ⋅ 3^9[122X, which divides [22X|M|[122X.
        So  [22X|U|  ≤ 2^8 ⋅ 3^9[122X, and a possible counterexample to the proposition
        must  satisfy [22X|N_G(U)| ≥ |G| / (2^8 ⋅ 3^9) = 811588377600[122X. We have [22X|M|
        =  29713078886400[122X,  which  is  less  than  [22X37[122X times this minimal order
        required  for  [22XN_G(U)[122X.  However, the intersection [22XH[122X of this group with
        the  simple  subgroup  [22XS ≅ O_8^+(3)[122X in [22XM[122X cannot be at most [22X36[122X, because
        the  largest  maximal  subgroups in [22XS[122X have index [22X1080[122X (see [CCN+85, p.
        140]).  Arguing  not  with  [22XS[122X  but with [22XM[122X, we can show –using only the
        character  table of [22XM[122X– that all proper subgroups of index less than [22X37
        ⋅ 6[122X in [22XM[122X contain [22XS[122X.[133X
  
  
  [1X7.2 [33X[0;0YThe Proof[133X[101X
  
  [33X[0;0YThe following [5XGAP[105X function utilizes Lemma 1. Its input are the [5XGAP[105X character
  table  [10Xtbl[110X  of  a  group [22XG[122X, say, and a list [10Xmaxesinfo[110X of character tables of
  maximal  subgroups of [22XG[122X, covering at least all those maximal subgroups [22XM[122X for
  which [22X|M|^2 ≥ |G|[122X holds.[133X
  
  [33X[0;0YThe  idea  is  to collect pairs [22X(M, g)[122X that satisfy part (a) of Lemma 1, and
  then to show that they do not satisfy part (b) or part (c). For each maximal
  subgroup  [22XM[122X that admits elements [22Xg[122X as in Lemma 1, information is printed how
  this candidate is excluded.[133X
  
  [33X[0;0YThe  function returns a list of length three. The first entry is [9Xtrue[109X if the
  criteria of Lemma 1 are sufficient to prove that the proposition is true for
  [22XG[122X,  and  [9Xfalse[109X  otherwise.  The second entry is the name of [22XG[122X, and the third
  entry  in  the  number  of  maximal subgroups [22XM[122X for which an element [22Xg[122X as in
  Lemma 1 (a) exists.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApplyTheLemma:= function( tbl, maxesinfo )[127X[104X
    [4X[25X>[125X [27X    local Gname, Gsize, cents, orders, result, Mtbl, Msize, maxc, i,[127X[104X
    [4X[25X>[125X [27X          pi, pipart, c, Mclasslengths, Fit, excluded, Kclasses, Mbar,[127X[104X
    [4X[25X>[125X [27X          Ksize, Sclasses, Ssize, d;[127X[104X
    [4X[25X>[125X [27X    Gname:= Identifier( tbl );[127X[104X
    [4X[25X>[125X [27X    Gsize:= Size( tbl );[127X[104X
    [4X[25X>[125X [27X    cents:= SizesCentralizers( tbl );[127X[104X
    [4X[25X>[125X [27X    orders:= OrdersClassRepresentatives( tbl );[127X[104X
    [4X[25X>[125X [27X    result:= [ true, Gname, 0 ];[127X[104X
    [4X[25X>[125X [27X    # Run over the relevant maximal subgroups.[127X[104X
    [4X[25X>[125X [27X    for Mtbl in maxesinfo do[127X[104X
    [4X[25X>[125X [27X      Msize:= Size( Mtbl );[127X[104X
    [4X[25X>[125X [27X      # Run over nonidentity class representatives g of squarefree[127X[104X
    [4X[25X>[125X [27X      # order, compute the largest c that occurs.[127X[104X
    [4X[25X>[125X [27X      maxc:= 1;[127X[104X
    [4X[25X>[125X [27X      for i in [ 2 .. NrConjugacyClasses( tbl ) ] do[127X[104X
    [4X[25X>[125X [27X        pi:= Factors( orders[i] );[127X[104X
    [4X[25X>[125X [27X        if IsSet( pi ) then[127X[104X
    [4X[25X>[125X [27X          # The elements in class `i' have squarefree order.[127X[104X
    [4X[25X>[125X [27X          pipart:= Product( Filtered( Factors( cents[i] ),[127X[104X
    [4X[25X>[125X [27X                                      x -> x in pi ) );[127X[104X
    [4X[25X>[125X [27X          c:= Gcd( pipart, Msize );[127X[104X
    [4X[25X>[125X [27X          if maxc < c then[127X[104X
    [4X[25X>[125X [27X            maxc:= c;[127X[104X
    [4X[25X>[125X [27X          fi;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X      if maxc * Msize >= Gsize then[127X[104X
    [4X[25X>[125X [27X        # Criterion (a) is satisfied, try to exclude (b) and (c).[127X[104X
    [4X[25X>[125X [27X        result[3]:= result[3] + 1;[127X[104X
    [4X[25X>[125X [27X        Print( Gname, ": consider M = ", Identifier( Mtbl ),[127X[104X
    [4X[25X>[125X [27X               ", c = ", StringPP( maxc ),[127X[104X
    [4X[25X>[125X [27X               ", c * |M| / |G| >= ", Int( maxc * Msize / Gsize ),[127X[104X
    [4X[25X>[125X [27X               "\n" );[127X[104X
    [4X[25X>[125X [27X        Mclasslengths:= SizesConjugacyClasses( Mtbl );[127X[104X
    [4X[25X>[125X [27X        Fit:= Mclasslengths{ ClassPositionsOfFittingSubgroup( Mtbl ) };[127X[104X
    [4X[25X>[125X [27X        if Sum( Fit ) * Msize >= Gsize then[127X[104X
    [4X[25X>[125X [27X          # Criterion (b1) is satisfied.[127X[104X
    [4X[25X>[125X [27X          Print( Gname, ": not excludable by (b1)\n" );[127X[104X
    [4X[25X>[125X [27X          result[1]:= false;[127X[104X
    [4X[25X>[125X [27X        elif maxc * Msize < 2 * Gsize then[127X[104X
    [4X[25X>[125X [27X          # Criterion (b2) is not satisfied.[127X[104X
    [4X[25X>[125X [27X          Print( Gname, ":     excluded by (b2)\n" );[127X[104X
    [4X[25X>[125X [27X        else[127X[104X
    [4X[25X>[125X [27X          # Run over the normal subgroups of M.[127X[104X
    [4X[25X>[125X [27X          excluded:= false;[127X[104X
    [4X[25X>[125X [27X          for Kclasses in ClassPositionsOfNormalSubgroups( Mtbl ) do[127X[104X
    [4X[25X>[125X [27X            Mbar:= Mtbl / Kclasses;[127X[104X
    [4X[25X>[125X [27X            Ksize:= Sum( Mclasslengths{ Kclasses } );[127X[104X
    [4X[25X>[125X [27X            if IsAlmostSimpleCharacterTable( Mbar ) and[127X[104X
    [4X[25X>[125X [27X               Ksize * Msize < Gsize then[127X[104X
    [4X[25X>[125X [27X              # We are in the situation of criterion (c).[127X[104X
    [4X[25X>[125X [27X              # The socle is the unique minimal normal subgroup.[127X[104X
    [4X[25X>[125X [27X              Sclasses:= ClassPositionsOfMinimalNormalSubgroups([127X[104X
    [4X[25X>[125X [27X                             Mbar )[1];[127X[104X
    [4X[25X>[125X [27X              Ssize:= Sum( SizesConjugacyClasses( Mbar ){ Sclasses } );[127X[104X
    [4X[25X>[125X [27X              d:= Int( maxc * Msize * Size( Mbar )[127X[104X
    [4X[25X>[125X [27X                  / ( Gsize * Ssize ) );[127X[104X
    [4X[25X>[125X [27X              # Try to show that all subgroups of index up to d[127X[104X
    [4X[25X>[125X [27X              # in Mbar contain the socle.[127X[104X
    [4X[25X>[125X [27X              if ForAll( [ 2 .. d ],[127X[104X
    [4X[25X>[125X [27X                   n -> ForAll( PermChars( Mbar, rec( torso:= [ n ] ) ),[127X[104X
    [4X[25X>[125X [27X                          chi -> IsSubset([127X[104X
    [4X[25X>[125X [27X                                     ClassPositionsOfKernel( chi ),[127X[104X
    [4X[25X>[125X [27X                                     Sclasses ) ) ) then[127X[104X
    [4X[25X>[125X [27X                Print( Gname, ":     excluded by (c), |K| = ",[127X[104X
    [4X[25X>[125X [27X                       StringPP( Ksize ), ", degree bound ", d, "\n" );[127X[104X
    [4X[25X>[125X [27X                excluded:= true;[127X[104X
    [4X[25X>[125X [27X                break;[127X[104X
    [4X[25X>[125X [27X              fi;[127X[104X
    [4X[25X>[125X [27X            fi;[127X[104X
    [4X[25X>[125X [27X          od;[127X[104X
    [4X[25X>[125X [27X          if not excluded then[127X[104X
    [4X[25X>[125X [27X            Print( Gname, ": not excludable by (c)\n" );[127X[104X
    [4X[25X>[125X [27X            result[1]:= false;[127X[104X
    [4X[25X>[125X [27X          fi;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    return result;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  our proof relies on the classifications of maximal subgroups of sporadic
  simple groups, see [CCN+85] and [BN95].[133X
  
  [33X[0;0YThe [5XGAP[105X Character Table Library [Bre22] contains the character tables of the
  sporadic  simple  groups and of their maximal subgroups, except that not all
  character  tables  of  maximal  subgroups of the Monster group are available
  yet. (See Section [14X7.1[114X for the treatment of the Monster group.)[133X
  
  [33X[0;0YSince  the  [5XGAP[105X Character Table Library is used for the computations in this
  section, we first load this package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "ctbllib", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we apply the function to the sporadic simple groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= [];;                                       [127X[104X
    [4X[25Xgap>[125X [27Xfor name in AllCharacterTableNames( IsSporadicSimple, true,[127X[104X
    [4X[25X>[125X [27X                                       IsDuplicateTable, false ) do[127X[104X
    [4X[25X>[125X [27X     tbl:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     if HasMaxes( tbl ) then[127X[104X
    [4X[25X>[125X [27X       mx:= List( Maxes( tbl ), CharacterTable );  [127X[104X
    [4X[25X>[125X [27X     elif name = "M" then[127X[104X
    [4X[25X>[125X [27X       mx:= [ CharacterTable( "2.B" ) ];[127X[104X
    [4X[25X>[125X [27X     else[127X[104X
    [4X[25X>[125X [27X       Error( "this should not happen ...");[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X     Add( info, ApplyTheLemma( tbl, mx ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28XB: consider M = 2.2E6(2).2, c = 2^38, c * |M| / |G| >= 20[128X[104X
    [4X[28XB:     excluded by (c), |K| = 2, degree bound 40[128X[104X
    [4X[28XCo1: consider M = Co2, c = 2^13*3^5, c * |M| / |G| >= 20[128X[104X
    [4X[28XCo1:     excluded by (c), |K| = 1, degree bound 20[128X[104X
    [4X[28XCo1: consider M = 3.Suz.2, c = 2^13*3^5, c * |M| / |G| >= 1[128X[104X
    [4X[28XCo1:     excluded by (b2)[128X[104X
    [4X[28XCo2: consider M = U6(2).2, c = 2^16, c * |M| / |G| >= 28[128X[104X
    [4X[28XCo2:     excluded by (c), |K| = 1, degree bound 56[128X[104X
    [4X[28XCo2: consider M = 2^10:m22:2, c = 2^18, c * |M| / |G| >= 5[128X[104X
    [4X[28XCo2:     excluded by (c), |K| = 2^10, degree bound 11[128X[104X
    [4X[28XCo2: consider M = 2^1+8:s6f2, c = 2^18, c * |M| / |G| >= 4[128X[104X
    [4X[28XCo2:     excluded by (c), |K| = 2^9, degree bound 4[128X[104X
    [4X[28XCo3: consider M = McL.2, c = 2^4*3^4, c * |M| / |G| >= 4[128X[104X
    [4X[28XCo3:     excluded by (c), |K| = 1, degree bound 9[128X[104X
    [4X[28XF3+: consider M = Fi23, c = 2^9*3^9, c * |M| / |G| >= 32[128X[104X
    [4X[28XF3+:     excluded by (c), |K| = 1, degree bound 32[128X[104X
    [4X[28XFi22: consider M = 2.U6(2), c = 2^7*3^6, c * |M| / |G| >= 26[128X[104X
    [4X[28XFi22:     excluded by (c), |K| = 2, degree bound 26[128X[104X
    [4X[28XFi22: consider M = O7(3), c = 2^7*3^6, c * |M| / |G| >= 6[128X[104X
    [4X[28XFi22:     excluded by (c), |K| = 1, degree bound 6[128X[104X
    [4X[28XFi22: consider M = Fi22M3, c = 2^7*3^6, c * |M| / |G| >= 6[128X[104X
    [4X[28XFi22:     excluded by (c), |K| = 1, degree bound 6[128X[104X
    [4X[28XFi22: consider M = O8+(2).3.2, c = 2^7*3^6, c * |M| / |G| >= 1[128X[104X
    [4X[28XFi22:     excluded by (b2)[128X[104X
    [4X[28XFi23: consider M = 2.Fi22, c = 2^8*3^9, c * |M| / |G| >= 159[128X[104X
    [4X[28XFi23:     excluded by (c), |K| = 2, degree bound 159[128X[104X
    [4X[28XFi23: consider M = O8+(3).3.2, c = 2^8*3^9, c * |M| / |G| >= 36[128X[104X
    [4X[28XFi23:     excluded by (c), |K| = 1, degree bound 219[128X[104X
    [4X[28XHS: consider M = M22, c = 2^7, c * |M| / |G| >= 1[128X[104X
    [4X[28XHS:     excluded by (b2)[128X[104X
    [4X[28XM11: consider M = A6.2_3, c = 2^4, c * |M| / |G| >= 1[128X[104X
    [4X[28XM11:     excluded by (b2)[128X[104X
    [4X[28XM12: consider M = M11, c = 2^4, c * |M| / |G| >= 1[128X[104X
    [4X[28XM12:     excluded by (b2)[128X[104X
    [4X[28XM12: consider M = M12M2, c = 2^4, c * |M| / |G| >= 1[128X[104X
    [4X[28XM12:     excluded by (b2)[128X[104X
    [4X[28XM22: consider M = L3(4), c = 2^6, c * |M| / |G| >= 2[128X[104X
    [4X[28XM22:     excluded by (c), |K| = 1, degree bound 2[128X[104X
    [4X[28XM22: consider M = 2^4:a6, c = 2^7, c * |M| / |G| >= 1[128X[104X
    [4X[28XM22:     excluded by (b2)[128X[104X
    [4X[28XM23: consider M = M22, c = 2^7, c * |M| / |G| >= 5[128X[104X
    [4X[28XM23:     excluded by (c), |K| = 1, degree bound 5[128X[104X
    [4X[28XM24: consider M = M23, c = 2^7, c * |M| / |G| >= 5[128X[104X
    [4X[28XM24:     excluded by (c), |K| = 1, degree bound 5[128X[104X
    [4X[28XM24: consider M = 2^4:a8, c = 2^10, c * |M| / |G| >= 1[128X[104X
    [4X[28XM24:     excluded by (b2)[128X[104X
    [4X[28XMcL: consider M = U4(3), c = 3^6, c * |M| / |G| >= 2[128X[104X
    [4X[28XMcL:     excluded by (c), |K| = 1, degree bound 2[128X[104X
    [4X[28XRu: consider M = 2F4(2)'.2, c = 2^12, c * |M| / |G| >= 1[128X[104X
    [4X[28XRu:     excluded by (b2)[128X[104X
    [4X[28XSuz: consider M = G2(4), c = 2^12, c * |M| / |G| >= 2[128X[104X
    [4X[28XSuz:     excluded by (c), |K| = 1, degree bound 2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  of  all,  we see that Lemma 1 is sufficient to prove the proposition,
  since all candidates were excluded.[133X
  
  [33X[0;0YMoreover,  we  see  that  for  the  following ten sporadic simple groups, no
  candidates  had  to  be  considered. (No information was printed about these
  groups.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFiltered( info, x -> x[3] = 0 );[127X[104X
    [4X[28X[ [ true, "HN", 0 ], [ true, "He", 0 ], [ true, "J1", 0 ], [128X[104X
    [4X[28X  [ true, "J2", 0 ], [ true, "J3", 0 ], [ true, "J4", 0 ], [128X[104X
    [4X[28X  [ true, "Ly", 0 ], [ true, "M", 0 ], [ true, "ON", 0 ], [128X[104X
    [4X[28X  [ true, "Th", 0 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X7.3 [33X[0;0YAlternative: Use [5XGAP[105X[101X[1X's Tables of Marks[133X[101X
  
  [33X[0;0YWe  can easily inspect all conjugacy classes of subgroups of a group [22XG[122X whose
  table  of  marks  is  contained in [5XGAP[105X's Library of Tables of Marks [NMP18].
  First we load this [5XGAP[105X package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "tomlib", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe following [5XGAP[105X function takes the table of marks of a group [22XG[122X and returns
  the  list  of  pairs  [22X[  U,  N_G(U) ][122X where [22XU[122X ranges over representatives of
  conjugacy classes of those nilpotent subgroups of [22XG[122X for which [22X|U| ⋅ |N_G(U)|[122X
  is maximal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaximalpairs:= function( tom )[127X[104X
    [4X[25X>[125X [27X   local g, max, result, i, u, n, prod;[127X[104X
    [4X[25X>[125X [27X   g:= UnderlyingGroup( tom );[127X[104X
    [4X[25X>[125X [27X   max:= 1;[127X[104X
    [4X[25X>[125X [27X   result:= [];[127X[104X
    [4X[25X>[125X [27X   for i in [ 1 .. Length( OrdersTom( tom ) ) ] do[127X[104X
    [4X[25X>[125X [27X     u:= RepresentativeTom( tom, i );[127X[104X
    [4X[25X>[125X [27X     if not IsTrivial( u ) and IsNilpotent( u ) then[127X[104X
    [4X[25X>[125X [27X       n:= Normalizer( g, u );[127X[104X
    [4X[25X>[125X [27X       prod:= Size( u ) * Size( n );[127X[104X
    [4X[25X>[125X [27X       if max < prod then[127X[104X
    [4X[25X>[125X [27X         max:= prod;[127X[104X
    [4X[25X>[125X [27X         result:= [ [ u, n ] ];[127X[104X
    [4X[25X>[125X [27X       elif max = prod then[127X[104X
    [4X[25X>[125X [27X         Add( result, [ u, n ] );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25X>[125X [27X   return result;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  let  us  collect the data for those sporadic simple groups for which the
  table of marks is known.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in AllCharacterTableNames( IsSporadicSimple, true,[127X[104X
    [4X[25X>[125X [27X                                       IsDuplicateTable, false ) do[127X[104X
    [4X[25X>[125X [27X     tom:= TableOfMarks( name );[127X[104X
    [4X[25X>[125X [27X     if tom <> fail then[127X[104X
    [4X[25X>[125X [27X       Add( info, [ name, tom, maximalpairs( tom ) ] );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XLength( info );[127X[104X
    [4X[28X12[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe got results for twelve sporadic simple groups. The following computations
  show  that  in  ten  cases,  the  simple  group [22XG[122X contains a unique class of
  nontrivial  nilpotent  subgroups  [22XU[122X  for  which  the  maximal value of [22X|U| ⋅
  |N_G(U)|[122X  is  attained.  The ratio of this value and [22X|G|[122X is less than [22X21[122X per
  cent. The following table shows the name of the group [22XG[122X, the orders of [22XU[122X and
  [22XN_G(U)[122X, and the integral part of [22X10^6[122X times the ratio.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( info, x -> Length( x[3] ) );[127X[104X
    [4X[28X[ 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xmat:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor entry in info do[127X[104X
    [4X[25X>[125X [27X     pair:= entry[3][1];                          # [ U, N_G(U) ][127X[104X
    [4X[25X>[125X [27X     bound:= Size( pair[1] ) * Size( pair[2] );   # |U|*|N_G(U)|[127X[104X
    [4X[25X>[125X [27X     size:= Size( UnderlyingGroup( entry[2] ) );  # |G|[127X[104X
    [4X[25X>[125X [27X     Add( mat, [ entry[1],[127X[104X
    [4X[25X>[125X [27X                 StringPP( Size( pair[1] ) ),[127X[104X
    [4X[25X>[125X [27X                 StringPP( Size( pair[2] ) ),[127X[104X
    [4X[25X>[125X [27X                 Int( 10^6 * bound / size ) ] );[127X[104X
    [4X[25X>[125X [27X     if Size( pair[1] ) * Size( pair[2] ) >= 21/100 * size then[127X[104X
    [4X[25X>[125X [27X       Error("!");[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XPrintArray( mat );[127X[104X
    [4X[28X[ [           Co3,           3^5,  2^5*3^7*5*11,          1886 ],[128X[104X
    [4X[28X  [            HS,           2^6,       2^9*3*7,         15515 ],[128X[104X
    [4X[28X  [            He,           2^6,    2^10*3^3*5,          2195 ],[128X[104X
    [4X[28X  [            J1,            19,        2*3*19,         12337 ],[128X[104X
    [4X[28X  [            J2,           2^6,       2^7*3^2,        121904 ],[128X[104X
    [4X[28X  [            J3,           3^5,       2^3*3^5,          9404 ],[128X[104X
    [4X[28X  [           M11,           3^2,       2^4*3^2,        163636 ],[128X[104X
    [4X[28X  [           M12,           2^5,         2^6*3,         64646 ],[128X[104X
    [4X[28X  [           M22,           2^4,     2^7*3^2*5,        207792 ],[128X[104X
    [4X[28X  [           M23,           2^4,   2^7*3^2*5*7,         63241 ],[128X[104X
    [4X[28X  [           M24,           2^6,    2^10*3^3*5,         36137 ],[128X[104X
    [4X[28X  [           McL,           3^5,     2^4*3^6*5,         15779 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YMoreover,  we  see  that in most cases, the group [22XU[122X for which the maximum is
  attained is not the largest [22Xp[122X-subgroup in the simple group in question.[133X
  
