<h3>6 <span class="Heading">Solvable Subgroups of Maximal Order in Sporadic Simple Groups</span></h3>
<p>Date: May 14th, 2012</p>
<p>We determine the orders of solvable subgroups of maximal orders in sporadic simple groups and their automorphism groups, using the information in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib_mj.html#biBCCN85">[CCN+85]</a> and the <strong class="pkg">GAP</strong> system <a href="chapBib_mj.html#biBGAP">[GAP24]</a>, in particular its Character Table Library <a href="chapBib_mj.html#biBCTblLib">[Bre25]</a> and its library of Tables of Marks <a href="chapBib_mj.html#biBTomLib">[MNP19]</a>.</p>
<p>We also determine the conjugacy classes of these solvable subgroups in the big group, and the maximal overgroups.</p>
<p>A first version of this document, which was based on <strong class="pkg">GAP</strong> 4.4.10, had been accessible in the web since August 2006. The differences to the current version are as follows.</p>
<ul>
<li><p>The format of the <strong class="pkg">GAP</strong> output was adjusted to the changed behaviour of <strong class="pkg">GAP</strong> 4.5.</p>
</li>
<li><p>The (too wide) table of results was split into two tables, the first one lists the orders and indices of the subgroups, the second one lists the structure of subgroups and the maximal overgroups.</p>
</li>
<li><p>The distribution of the solvable subgroups of maximal orders in the Baby Monster group and the Monster group to conjugacy classes is now proved.</p>
</li>
<li><p>The sporadic simple Monster group has exactly one class of maximal subgroups of the type PSL<span class="SimpleMath">\((2, 41)\)</span> (see <a href="chapBib_mj.html#biBNW12">[NW13]</a>), and has no maximal subgroups which have the socle PSL<span class="SimpleMath">\((2, 27)\)</span> (see <a href="chapBib_mj.html#biBWil10">[Wil10]</a>). This does not affect the arguments in Section <a href="chap6_mj.html#X87D468D07D7237CB"><span class="RefLink">6.4-14</span></a>, but some statements in this section had to be corrected.</p>
<p>The tables I and II list information about solvable subgroups of maximal order in sporadic simple groups and their automorphism groups. The first column in each table gives the names of the almost simple groups <span class="SimpleMath">\(G\)</span>, in alphabetical order. The remaining columns of Table I contain the order and the index of a solvable subgroup <span class="SimpleMath">\(S\)</span> of maximal order in <span class="SimpleMath">\(G\)</span>, the value <span class="SimpleMath">\(\log_{|G|}(|S|)\)</span>, and the page number in the <strong class="pkg">Atlas</strong> <a href="chapBib_mj.html#biBCCN85">[CCN+85]</a> where the information about maximal subgroups of <span class="SimpleMath">\(G\)</span> is listed. The second and third columns of Table II show a structure description of <span class="SimpleMath">\(S\)</span> and the structures of the maximal subgroups that contain <span class="SimpleMath">\(S\)</span>; the value <q><span class="SimpleMath">\(S\)</span></q> in the third column means that <span class="SimpleMath">\(S\)</span> is itself maximal in <span class="SimpleMath">\(G\)</span>. The fourth and fifth columns list the pages in the <strong class="pkg">Atlas</strong> with the information about the maximal subgroups of <span class="SimpleMath">\(G\)</span> and the section in this note with the proof of the table row, respectively. In the fourth column, page numbers in brackets refer to the <strong class="pkg">Atlas</strong> pages with information about the maximal subgroups of nonsolvable quotients of the maximal subgroups of <span class="SimpleMath">\(G\)</span> listed in the third column.</p>
<p>Note that in the case of nonmaximal subgroups <span class="SimpleMath">\(S\)</span>, we do not claim to describe the <em>module</em> structure of <span class="SimpleMath">\(S\)</span> in the third column of the table; we have kept the <strong class="pkg">Atlas</strong> description of the normal subgroups of the maximal overgroups of <span class="SimpleMath">\(S\)</span>. For example, the subgroup <span class="SimpleMath">\(S\)</span> listed for <span class="SimpleMath">\(Co_2\)</span> is contained in maximal subgroups of the types <span class="SimpleMath">\(2^{1+8}_+:S_6(2)\)</span> and <span class="SimpleMath">\(2^{4+10}(S_4 \times S_3)\)</span>, so <span class="SimpleMath">\(S\)</span> has normal subgroups of the orders <span class="SimpleMath">\(2\)</span>, <span class="SimpleMath">\(2^4\)</span>, <span class="SimpleMath">\(2^9\)</span>, <span class="SimpleMath">\(2^{14}\)</span>, and <span class="SimpleMath">\(2^{16}\)</span>; more <strong class="pkg">Atlas</strong> conformal notations would be <span class="SimpleMath">\(2^{[14]}(S_4 \times S_3)\)</span> or <span class="SimpleMath">\(2^{[16]}(S_3 \times S_3)\)</span>.</p>
<p>As a corollary (see Section <a href="chap6_mj.html#X7CD8E04C7F32AD56"><span class="RefLink">6.5</span></a>), we read off the following.</p>
<p>Corollary:</p>
<p>Exactly the following almost simple groups <span class="SimpleMath">\(G\)</span> with sporadic simple socle contain a solvable subgroup <span class="SimpleMath">\(S\)</span> with the property <span class="SimpleMath">\(|S|^2 \geq |G|\)</span>.</p>
<p>The existence of the subgroups <span class="SimpleMath">\(S\)</span> of <span class="SimpleMath">\(G\)</span> with the structure and the order stated in Table I and II follows from the <strong class="pkg">Atlas</strong>: It is obvious in the cases where <span class="SimpleMath">\(S\)</span> is maximal in <span class="SimpleMath">\(G\)</span>, and in the other cases, the <strong class="pkg">Atlas</strong> information about a nonsolvable factor group of a maximal subgroup of <span class="SimpleMath">\(G\)</span> suffices.</p>
<p>In order to show that the table rows for the group <span class="SimpleMath">\(G\)</span> are correct, we have to show the following.</p>
<ul>
<li><p><span class="SimpleMath">\(G\)</span> does not contain solvable subgroups of order larger than <span class="SimpleMath">\(|S|\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(G\)</span> contain exactly the conjugacy classes of solvable subgroups of order <span class="SimpleMath">\(|S|\)</span> that are listed in the second column of Table II.</p>
</li>
<li><p><span class="SimpleMath">\(S\)</span> is contained exactly in the maximal subgroups listed in the third column of Table II.</p>
</li>
</ul>
<p><em>Remark:</em></p>
<ul>
<li><p>Each of the groups <span class="SimpleMath">\(M_{12}\)</span> and <span class="SimpleMath">\(He\)</span> contains two classes of isomorphic solvable subgroups of maximal order.</p>
</li>
<li><p>Each of the groups <span class="SimpleMath">\(Ru\)</span>, <span class="SimpleMath">\(Th\)</span>, and <span class="SimpleMath">\(M\)</span> contains two classes of nonisomorphic solvable subgroups of maximal order.</p>
</li>
<li><p>The solvable subgroups of maximal order in <span class="SimpleMath">\(McL.2\)</span> have the structure <span class="SimpleMath">\(3^{1+4}_+:4S_4\)</span>, the subgroups are maximal in the maximal subgroups of the structures <span class="SimpleMath">\(3^{1+4}_+:4S_5\)</span> and <span class="SimpleMath">\(U_4(3).2_3\)</span> in <span class="SimpleMath">\(McL.2\)</span>. Note that the <strong class="pkg">Atlas</strong> claims another structure for these maximal subgroups of <span class="SimpleMath">\(U_4(3).2_3\)</span>, see <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. 52]</a>.</p>
</li>
<li><p>The solvable subgroups of maximal order in <span class="SimpleMath">\(Co_3\)</span> are the normalizers of Sylow <span class="SimpleMath">\(3\)</span>-subgroups of <span class="SimpleMath">\(Co_3\)</span>.</p>
<p>We combine the information in the <strong class="pkg">Atlas</strong> <a href="chapBib_mj.html#biBCCN85">[CCN+85]</a> with explicit computations using the <strong class="pkg">GAP</strong> system <a href="chapBib_mj.html#biBGAP">[GAP24]</a>, in particular its Character Table Library <a href="chapBib_mj.html#biBCTblLib">[Bre25]</a> and its library of Tables of Marks <a href="chapBib_mj.html#biBTomLib">[MNP19]</a>. First we load these two packages.</p>
<h5>6.2-1 <span class="Heading">Use the Table of Marks</span></h5>
<p>If the <strong class="pkg">GAP</strong> library of Tables of Marks <a href="chapBib_mj.html#biBTomLib">[MNP19]</a> contains the table of marks of a group <span class="SimpleMath">\(G\)</span> then we can easily inspect all conjugacy classes of subgroups of <span class="SimpleMath">\(G\)</span>. The following small <strong class="pkg">GAP</strong> function can be used for that. It returns <code class="keyw">false</code> if the table of marks of the group with the name <code class="code">name</code> is not available, and the list <code class="code">[ name, n, super ]</code> otherwise, where <code class="code">n</code> is the maximal order of solvable subgroups of <span class="SimpleMath">\(G\)</span>, and <code class="code">super</code> is a list of lists; for each conjugacy class of solvable subgroups <span class="SimpleMath">\(S\)</span> of order <code class="code">n</code>, <code class="code">super</code> contains the list of orders of representatives <span class="SimpleMath">\(M\)</span> of the classes of maximal subgroups of <span class="SimpleMath">\(G\)</span> such that <span class="SimpleMath">\(M\)</span> contains a conjugate of <span class="SimpleMath">\(S\)</span>.</p>
<p>Note that a subgroup in the <span class="SimpleMath">\(i\)</span>-th class of a table of marks contains a subgroup in the <span class="SimpleMath">\(j\)</span>-th class if and only if the entry in the position <span class="SimpleMath">\((i,j)\)</span> of the table of marks is nonzero. For tables of marks objects in <strong class="pkg">GAP</strong>, this is the case if and only if <span class="SimpleMath">\(j\)</span> is contained in the <span class="SimpleMath">\(i\)</span>-th row of the list that is stored as the value of the attribute <code class="code">SubsTom</code> of the table of marks object; for this test, one need not unpack the matrix of marks.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSolvableSubgroupInfoFromTom:= function( name )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> local tom, # table of marks for `name'
<span class="GAPprompt">></span> <span class="GAPinput"> n, # maximal order of a solvable subgroup</span>
<span class="GAPprompt">></span> <span class="GAPinput"> maxsubs, # numbers of the classes of subgroups of order `n'
<span class="GAPprompt">></span> <span class="GAPinput"> orders, # list of orders of the classes of subgroups</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i, # loop over the classes of subgroups</span>
<span class="GAPprompt">></span> <span class="GAPinput"> maxes, # list of positions of the classes of max. subgroups</span>
<span class="GAPprompt">></span> <span class="GAPinput"> subs, # `SubsTom' value
<span class="GAPprompt">></span> <span class="GAPinput"> cont; # list of list of positions of max. subgroups</span>
<span class="GAPprompt">></span> <span class="GAPinput"></span>
<span class="GAPprompt">></span> <span class="GAPinput"> tom:= TableOfMarks( name );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if tom = fail then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return false;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n:= 1;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> maxsubs:= [];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> orders:= OrdersTom( tom );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> for i in [ 1 .. Length( orders ) ] do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if IsSolvableTom( tom, i ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if orders[i] = n then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Add( maxsubs, i );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> elif orders[i] > n then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> n:= orders[i];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> maxsubs:= [ i ];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> maxes:= MaximalSubgroupsTom( tom )[1];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> subs:= SubsTom( tom );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> cont:= List( maxsubs, j -> Filtered( maxes, i -> j in subs[i] ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"></span>
<span class="GAPprompt">></span> <span class="GAPinput"> return [ name, n, List( cont, l -> orders{ l } ) ];</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
</pre></div>
<h5>6.2-2 <span class="Heading">Use Information from the Character Table Library</span></h5>
<p>The <strong class="pkg">GAP</strong> Character Table Library contains the character tables of all maximal subgroups of sporadic simple groups, except for the Monster group. This information can be used as follows.</p>
<p>We start, for a sporadic simple group <span class="SimpleMath">\(G\)</span>, with a known solvable subgroup of order <span class="SimpleMath">\(n\)</span>, say, in <span class="SimpleMath">\(G\)</span>. In order to show that <span class="SimpleMath">\(G\)</span> contains no solvable subgroup of larger order, it suffices to show that no maximal subgroup of <span class="SimpleMath">\(G\)</span> contains a larger solvable subgroup.</p>
<p>The point is that usually the orders of the maximal subgroups of <span class="SimpleMath">\(G\)</span> are not much larger than <span class="SimpleMath">\(n\)</span>, and that a maximal subgroup <span class="SimpleMath">\(M\)</span> contains a solvable subgroup of order <span class="SimpleMath">\(n\)</span> only if the factor group of <span class="SimpleMath">\(M\)</span> by its largest solvable normal subgroup <span class="SimpleMath">\(N\)</span> contains a solvable subgroup of order <span class="SimpleMath">\(n/|N|\)</span>. This reduces the question to relatively small groups.</p>
<p>What we can check <em>automatically</em> from the character table of <span class="SimpleMath">\(M/N\)</span> is whether <span class="SimpleMath">\(M/N\)</span> can contain subgroups (solvable or not) of indices between five and <span class="SimpleMath">\(|M|/n\)</span>, by computing possible permutation characters of these degrees. (Note that a solvable subgroup of a nonsolvable group has index at least five. This lower bound could be improved for example by considering the smallest degree of a nontrivial character, but this is not an issue here.)</p>
<p>Then we are left with a –hopefully short– list of maximal subgroups of <span class="SimpleMath">\(G\)</span>, together with upper bounds on the indices of possible solvable subgroups; excluding these possibilities then yields that the initially chosen solvable subgroup of <span class="SimpleMath">\(G\)</span> is indeed the largest one.</p>
<p>The following <strong class="pkg">GAP</strong> function can be used to compute this information for the character table <code class="code">tblM</code> of <span class="SimpleMath">\(M\)</span> and a given order <code class="code">minorder</code>. It returns <code class="keyw">false</code> if <span class="SimpleMath">\(M\)</span> cannot contain a solvable subgroup of order at least <code class="code">minorder</code>, otherwise a list <code class="code">[ tblM, m, k ]</code> where <code class="code">m</code> is the maximal index of a subgroup that has order at least <code class="code">minorder</code>, and <code class="code">k</code> is the minimal index of a possible subgroup of <span class="SimpleMath">\(M\)</span> (a proper subgroup if <span class="SimpleMath">\(M\)</span> is nonsolvable), according to the <strong class="pkg">GAP</strong> function <code class="func">PermChars</code> (<a href="../../../doc/ref/chap72_mj.html#X7D02541482C196A6"><span class="RefLink">Reference: PermChars</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SolvableSubgroupInfoFromCharacterTable:= function( tblM, minorder )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> local maxindex, # index of subgroups of order `minorder'
<span class="GAPprompt">></span> <span class="GAPinput"> N, # class positions describing a solvable normal subgroup</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fact, # character table of the factor by `N'
<span class="GAPprompt">></span> <span class="GAPinput"> classes, # class sizes in `fact'
<span class="GAPprompt">></span> <span class="GAPinput"> nsg, # list of class positions of normal subgroups</span>
<span class="GAPprompt">></span> <span class="GAPinput"> i; # loop over the possible indices</span>
<span class="GAPprompt">></span> <span class="GAPinput"></span>
<span class="GAPprompt">></span> <span class="GAPinput"> maxindex:= Int( Size( tblM ) / minorder );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if maxindex = 0 then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return false;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> elif IsSolvableCharacterTable( tblM ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return [ tblM, maxindex, 1 ];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> elif maxindex < 5 then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return false;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"></span>
<span class="GAPprompt">></span> <span class="GAPinput"> N:= [ 1 ];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fact:= tblM;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> repeat</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fact:= fact / N;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> classes:= SizesConjugacyClasses( fact );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> nsg:= Difference( ClassPositionsOfNormalSubgroups( fact ), [ [ 1 ] ] );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> N:= First( nsg, x -> IsPrimePowerInt( Sum( classes{ x } ) ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> until N = fail;</span>
<span class="GAPprompt">></span> <span class="GAPinput"></span>
<span class="GAPprompt">></span> <span class="GAPinput"> for i in Filtered( DivisorsInt( Size( fact ) ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> d -> 5 <= d and d <= maxindex ) do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if Length( PermChars( fact, rec( torso:= [ i ] ) ) ) > 0 then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return [ tblM, maxindex, i ];</span>
<span class="GAPprompt">></span> <span class="GAPinput"> fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"></span>
<span class="GAPprompt">></span> <span class="GAPinput"> return false;</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;;</span>
</pre></div>
<h4>6.3 <span class="Heading">Cases where the Table of Marks is available in <strong class="pkg">GAP</strong></span></h4>
<p>For twelve sporadic simple groups, the <strong class="pkg">GAP</strong> library of Tables of Marks knows the tables of marks, so we can use <code class="code">MaximalSolvableSubgroupInfoFromTom</code>.</p>
<p>We see that for <span class="SimpleMath">\(J_1\)</span>, <span class="SimpleMath">\(J_2\)</span>, <span class="SimpleMath">\(J_3\)</span>, <span class="SimpleMath">\(M_{11}\)</span>, and <span class="SimpleMath">\(M_{12}\)</span>, the subgroup <span class="SimpleMath">\(S\)</span> is maximal. For <span class="SimpleMath">\(M_{12}\)</span> and <span class="SimpleMath">\(He\)</span>, there are two classes of subgroups <span class="SimpleMath">\(S\)</span>. For the other groups, the class of subgroups <span class="SimpleMath">\(S\)</span> is unique, and there are one or two classes of maximal subgroups of <span class="SimpleMath">\(G\)</span> that contain <span class="SimpleMath">\(S\)</span>. From the shown orders of these maximal subgroups, their structures can be read off from the <strong class="pkg">Atlas</strong>, on the pages listed in Table II.</p>
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