| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/ctbllib/htm/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 22.4.2025 mit Größe 156 kB |
|
Quelle multfre2.htm Sprache: HTML |
|||||||||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd"> <html> <meta name="GENERATOR" content="TtH 3.59"> <style type="text/css"> div.p { margin-top: 7pt;}</style> <style type="text/css"><!-- td div.comp { margin-top: -0.6ex; margin-bottom: -1ex;} td div.comb { margin-top: -0.6ex; margin-bottom: -.6ex;} td div.hrcomp { line-height: 0.9; margin-top: -0.8ex; margin-bottom: -1ex;} td div.norm {line-height:normal;} span.roman {font-family: serif; font-style: normal; font-weight: normal;} span.overacc2 {position: relative; left: .8em; top: -1.2ex;} span.overacc1 {position: relative; left: .6em; top: -1.2ex;} --> <title> Multiplicity-Free Permutation Characters in GAP, part 2</title> <h1 align="center">Multiplicity-Free Permutation Characters in GAP, part 2 </h1> <body bgcolor="FFFFFF"> <div class="p"><!----></div> <h3 align="center"> T<font size="-2">HOMAS</font> B<font size="-2">REUER</font> <br /> <i>Lehrstuhl D für Mathematik</i> <br /> <i>RWTH, 52056 Aachen, Germany</i> </h3> <div class="p"><!----></div> <h3 align="center">May 30th, 2006 </h3> <div class="p"><!----></div> <div class="p"><!----></div> We complete the classification of the multiplicity-free permutation actions of nearly simple groups that involve a sporadic simple group, which had been started in [<a href="#BL96" name="CITEBL96">BL96</a>] and [<a href="#LM03" name=" <div class="p"><!----></div> <div class="p"><!----></div> <h1>Contents </h1><a href="#tth_sEc1" >1 Introduction</a><br /><a href="#tth_sEc2" >2 The Approach</a><br /> <a href="#tth_sEc2.1" >2.1 Computing Possible Permutation Characters</a><br /> <a href="#tth_sEc2.2" >2.2 Verifying the Candidates</a><br /> <a href="#tth_sEc2.3" >2.3 Isoclinic Groups</a><br /> <a href="#tth_sEc2.4" >2.4 Tests for <font face="helvetica">GAP</font></a><br /><a href="#tth_sEc3" >3 The Groups</a><br /> <a href="#tth_sEc3.1" >3.1 G = 2.M<sub>12</sub></a><br /> <a href="#tth_sEc3.2" >3.2 G = 2.M<sub>12</sub>.2</a><br /> <a href="#tth_sEc3.3" >3.3 G = 2.M<sub>22</sub></a><br /> <a href="#tth_sEc3.4" >3.4 G = 2.M<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.5" >3.5 G = 3.M<sub>22</sub></a><br /> <a href="#tth_sEc3.6" >3.6 G = 3.M<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.7" >3.7 G = 4.M<sub>22</sub> and G = 12.M<sub>22</sub></a><br /> <a href="#tth_sEc3.8" >3.8 G = 4.M<sub>22</sub>.2 and G = 12.M<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.9" >3.9 G = 6.M<sub>22</sub></a><br /> <a href="#tth_sEc3.10" >3.10 G = 6.M<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.11" >3.11 G = 2.J<sub>2</sub></a><br /> <a href="#tth_sEc3.12" >3.12 G = 2.J<sub>2</sub>.2</a><br /> <a href="#tth_sEc3.13" >3.13 G = 2.HS</a><br /> <a href="#tth_sEc3.14" >3.14 G = 2.HS.2</a><br /> <a href="#tth_sEc3.15" >3.15 G = 3.J<sub>3</sub></a><br /> <a href="#tth_sEc3.16" >3.16 G = 3.J<sub>3</sub>.2</a><br /> <a href="#tth_sEc3.17" >3.17 G = 3.McL</a><br /> <a href="#tth_sEc3.18" >3.18 G = 3.McL.2</a><br /> <a href="#tth_sEc3.19" >3.19 G = 2.Ru</a><br /> <a href="#tth_sEc3.20" >3.20 G = 2.Suz</a><br /> <a href="#tth_sEc3.21" >3.21 G = 2.Suz.2</a><br /> <a href="#tth_sEc3.22" >3.22 G = 3.Suz</a><br /> <a href="#tth_sEc3.23" >3.23 G = 3.Suz.2</a><br /> <a href="#tth_sEc3.24" >3.24 G = 6.Suz</a><br /> <a href="#tth_sEc3.25" >3.25 G = 6.Suz.2</a><br /> <a href="#tth_sEc3.26" >3.26 G = 3.ON</a><br /> <a href="#tth_sEc3.27" >3.27 G = 3.ON.2</a><br /> <a href="#tth_sEc3.28" >3.28 G = 2.Fi<sub>22</sub></a><br /> <a href="#tth_sEc3.29" >3.29 G = 2.Fi<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.30" >3.30 G = 3.Fi<sub>22</sub></a><br /> <a href="#tth_sEc3.31" >3.31 G = 3.Fi<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.32" >3.32 G = 6.Fi<sub>22</sub></a><br /> <a href="#tth_sEc3.33" >3.33 G = 6.Fi<sub>22</sub>.2</a><br /> <a href="#tth_sEc3.34" >3.34 G = 2.Co<sub>1</sub></a><br /> <a href="#tth_sEc3.35" >3.35 G = 3.F<sub>3+</sub></a><br /> <a href="#tth_sEc3.36" >3.36 G = 3.F<sub>3+</sub>.2</a><br /> <a href="#tth_sEc3.37" >3.37 G = 2.B</a><br /><a href="#tth_sEc4" >4 Appendix: Explicit Computations with Groups</a><br /> <a href="#tth_sEc4.1" >4.1 2<sup>4</sup>:A<sub>6</sub> type subgroups in 2.M<sub>22</sub></a><br /> <a href="#tth_sEc4.2" >4.2 2<sup>4</sup>:S<sub>5</sub> type subgroups in M<sub>22</sub>.2</a><br /> <a href="#tth_sEc4.3" >4.3 Multiplicities of Multiplicity-Free Actions of 6.Fi<sub>22</sub>.2</a><br /> <div class="p"><!----></div> <div class="p"><!----></div> <h2><a name="tth_sEc1"> 1</a> Introduction</h2> <div class="p"><!----></div> In [<a href="#BL96" name="CITEBL96">BL96</a>], the multiplicity-free permutation characters simple groups and their automorphism groups were classified. Based on this list, the multiplicity-free permutation characters of the central extensions of the sporadic simple groups were classified in [<a href="#LM03" name="CITELM03">LM</a>]. <div class="p"><!----></div> The purpose of this writeup is to show how the multiplicity-free permutation characters of the automorphic extensions of the central extensions of the sporadic simple groups can be computed, to verify the calculations in [<a href="#LM03" name="CITELM03">LM</a>] (and to correct an error, see Section <a href="#LMerror">3.32</a>), and to provide a test file for the <font face="helvetica">GAP</font> functions and the database. <div class="p"><!----></div> The database has been extended in the sense that also most of the character tables of the multiplicity-free permutation modules of the sporadic simple groups and their automorphic and central extensions have been computed, see [<a href="#Hoe01" name="CITEHoe01">Höh01</a>,<a href="#Mue03" name="CITEMue03">Mül03</a>,<a h <div class="p"><!----></div> Five errors in an earlier version (from July 2003) have been pointed out by Jürgen Müller. These errors concern the numbers of conjugacy classes of certain point stabilizers in 2.J<sub>2</sub>.2, 2.HS.2, and 6.Fi<sub>22</sub>.2 (see Sections <a href="#sect2J22">3.12</a>, <a href="#sect2HS2">3.14</a>, and <a href="#6Fi222">3. <div class="p"><!----></div> The only differences between the current version and the version that was available since 2005 are additions of references, adjustments of group names in the data file, and adjustments of the <font face="helvetica">GAP</font> output format to version 4.5, see [<a href="#GAP" name="CITEGAP">GAP24</a>]. <div class="p"><!----></div> Note that the version from 2003 was based on a data file that contained only the permutation character information, whereas the current version uses the database file of [<a href="#BM05" name="CITEBM05">BM05</a>], which includes also the known character tables of endomorphism rings. <div class="p"><!----></div> <h2><a name="tth_sEc2"> 2</a> The Approach</h2> <div class="p"><!----></div> Suppose that a group G contains a normal subgroup N. If π is a faithful multiplicity-free permutation character of G then π = 1<sub>U</sub><sup>G</sup> for a subgroup U of G that intersects N trivially, so π contains a constituent 1<sub>UN</sub><sup>G</sup> of degree π(1) / |N|, which can be viewed as a multiplicity-free permutation character of the factor group G / N. Moreover, no constituent of the difference π− 1<sub>UN</sub><sup>G</sup> has N in its kernel. <div class="p"><!----></div> So if we know all multiplicity-free permutation characters of the factor group G / N then we can compute all candidates for multiplicity-free permutation characters of G by "filling up" each such character [π] with a linear combination of characters not containing N in their kernels, of total degree (|N|−1) ·π(1), and such that the sum is a possible permutation character of G. For this situation, <font face="helvetica">GAP</font> provides a special variant of the functio <tt>PermChars</tt>. In a second step, the candidates are inspected whether the required point stabilizers (and if yes, how many conjugacy classes of them) exist. Finally, the permutation characters are verified by explicit induction from the character tables of the point stabilizers. <div class="p"><!----></div> The multiplicity-free permutation actions of the sporadic simple groups and their automorphism groups are known by [<a href="#BL96" name="CITEBL96">BL96</a>], so this approach is suitable for these groups. <div class="p"><!----></div> For central extensions of sporadic simple groups, the multiplicity-free permutation characters have been classified in [<a href="#LM03" name="CITELM03">LM</a>]; this note describes a slightly different approach, so we will give an independent confirmation of their results (except for the error pointed out in Section <a href="#LMerror">3.32</a>). <div class="p"><!----></div> First we load the Character Table Library [<a href="#CTblLib" name="CITECTblLib">Bre25</a>] of the <font face="helvetica">GAP</font> system [<a href="#GAP" name="CITEGAP">GAP24</a>], and the <font face="helvetica">GAP</font> interface (see [<a href="#AtlasRep" name="CITEAtlas A<font size="-2">TLAS</font> of Group Representations (see [<a href="#AGR" name="CITEAGR">WWT<s <div class="p"><!----></div> <pre> gap> LoadPackage( "ctbllib", false ); true gap> LoadPackage( "atlasrep", false ); true </pre> <div class="p"><!----></div> Then we read -if necessary- the file with <font face="helvetica">GAP</font> functions for comput multiplicity-free permutation characters, and the file with the data. Note that this includes the data we are going to compute, but we will actually <b>use</b> only the data for sporadic simple groups and their automorphism groups. For the other groups, we will compare the results computed below with the database. <div class="p"><!----></div> <pre> gap> if not IsBound( PossiblePermutationCharactersWithBoundedMultiplicity ) > then > ReadPackage( "ctbllib", "tst/multfree.g" ); > fi; gap> if not IsBound( MultFreeEndoRingCharacterTables ) then > ReadPackage( "ctbllib", "tst/mferctbl.gap" ); --------------------------------------------------------------------------- Loading the database of character tables of endomorphism rings of multiplicity-free permutation modules of the sporadic simple groups and their cyclic and bicyclic extensions, compiled by T. Breuer and J. Mueller. --------------------------------------------------------------------------- > fi; gap> if not IsBound( PossiblePermutationCharactersWithBoundedMultiplicity ) or > not IsBound( MultFreeEndoRingCharacterTables ) then > Print( "Sorry, the data files are not available!\n" ); > fi; </pre> <div class="p"><!----></div> (If the data files are not available then they can be fetched from the homepage of the <font face="helvetica">GAP</font> Character Table Library [<a href="#CTblLib" n <div class="p"><!----></div> <h3><a name="tth_sEc2.1"> 2.1</a> Computing Possible Permutation Characters</h3> <div class="p"><!----></div> Next we define the <font face="helvetica">GAP</font> functions that are needed in the following. <div class="p"><!----></div> The utility function <tt>PossiblePermutationCharacters</tt> takes two ordinary character tables <tt>sub</tt> and <tt>tbl</tt>, and returns the set of all induced class functions of the trivial character of <tt>sub</tt> to <tt>tbl</tt>, w.r.t. the possible class fusions from <tt>sub</tt> to <tt>tbl</tt>. (The entries in the result list are not necessarily multiplicity-free.) <div class="p"><!----></div> <div class="p"><!----></div> <pre> gap> PossiblePermutationCharacters:= function( sub, tbl ) > local fus, triv; > > fus:= PossibleClassFusions( sub, tbl ); > if fus = fail then > return fail; > fi; > triv:= [ TrivialCharacter( sub ) ]; > > return Set( List( fus, map -> Induced( sub, tbl, triv, map )[1] ) ); > end;; </pre> <div class="p"><!----></div> <tt>FaithfulCandidates</tt> takes the character table <tt>tbl</tt> of a group G and the name <tt>factname</tt> of a factor group F of G for which the multiplicity-free permutation characters are known, and returns a list of lists, the entry at the i-th position being the list of possible permutation characters of G that are multiplicity-free and such that the sum of all constituents that are characters of F is the i-th multiplicity-free permutation character of F. As a side-effect, if the i-th entry is nonempty then information is printed about the structure of the point-stabilizer in F and the number of candidates found. <div class="p"><!----></div> <pre> gap> FaithfulCandidates:= function( tbl, factname ) > local factinfo, factchars, facttbl, fus, sizeN, faith, i; > > # Fetch the data for the factor group. > factinfo:= MultFreeEndoRingCharacterTables( factname ); > factchars:= List( factinfo, x -> x.character ); > facttbl:= UnderlyingCharacterTable( factchars[1] ); > fus:= GetFusionMap( tbl, facttbl ); > sizeN:= Size( tbl ) / Size( facttbl ); > > # Compute faithful possible permutation characters. > faith:= List( factchars, pi -> PermChars( tbl, > rec( torso:= [ sizeN * pi[1] ], > normalsubgroup:= ClassPositionsOfKernel( fus ), > nonfaithful:= pi{ fus } ) ) ); > > # Take only the multiplicity-free ones. > faith:= List( faith, x -> Filtered( x, pi -> ForAll( Irr( tbl ), > chi -> ScalarProduct( tbl, pi, chi ) < 2 ) ) ); > > # Print info about the candidates. > for i in [ 1 .. Length( faith ) ] do > if not IsEmpty( faith[i] ) then > Print( i, ": subgroup ", factinfo[i].subgroup, > ", degree ", faith[i][1][1], > " (", Length( faith[i] ), " cand.)\n" ); > fi; > od; > > # Return the candidates. > return faith; > end;; </pre> <div class="p"><!----></div> <h3><a name="tth_sEc2.2"> 2.2</a> Verifying the Candidates</h3> <div class="p"><!----></div> In the verification step, we check which of the given candidates of G are induced from a given subgroup S. For that, we use the following function. Its arguments are the character table <tt>s</tt> of S, the character tables <tt>tbl2</tt> and <tt>tbl</tt> of G and its derived subgroup G<sup>′</sup> of index 2 (if G is perfect then <tt>0</tt> must be entered for <tt>tbl2</tt>), the list <tt>candidates</tt> of characters of G, and one of the strings <tt>"all"</tt>, <tt>"extending"</tt>, which means that we consider either all possible class fusions of <tt>s</tt> into <tt>tbl2</tt> or only those whose image does not lie in G<sup>′</sup>. Note that the table of the derived subgroup of G is needed because we want to express the decomposition of the permutation characters relative to G<sup>′</sup>. <div class="p"><!----></div> The idea is that we know that n different permutation characters arise from subgroups isomorphic with S (with the additional property that the image of the embedding of S into G is not contained in G<sup>′</sup> if the last argument is <tt>"extending"</tt>), and that <tt>candidates</tt> is a set of possible permutation characters, of length n. If the possible fusions between the character tables <tt>s</tt> and <tt>tbl2</tt> lead to exactly the given n permutation characters then we have proved that they are in fact the permutation characters of G in question. In this case, <tt>VerifyCandidates</tt> prints information about the decomposition of the permutation characters. If none of <tt>candidates</tt> arises from the possible embeddings of S into G then the function prints that S does not occur. In all other cases, the function signals an error. (This will not happen in the calls to this function below). <div class="p"><!----></div> <pre> gap> VerifyCandidates:= function( s, tbl, tbl2, candidates, admissible ) > local fus, der, pi; > > if tbl2 = 0 then > tbl2:= tbl; > fi; > > # Compute the possible class fusions, and induce the trivial character. > fus:= PossibleClassFusions( s, tbl2 ); > if admissible = "extending" then > der:= Set( GetFusionMap( tbl, tbl2 ) ); > fus:= Filtered( fus, map -> not IsSubset( der, map ) ); > fi; > pi:= Set( List( fus, map -> Induced( s, tbl2, > [ TrivialCharacter( s ) ], map )[1] ) ); > > # Compare the two lists. > if pi = SortedList( candidates ) then > Print( "G = ", Identifier( tbl2 ), ": point stabilizer ", > Identifier( s ), ", ranks ", > List( pi, x -> Length( ConstituentsOfCharacter(x) ) ), "\n" ); > if Size( tbl ) = Size( tbl2 ) then > Print( PermCharInfo( tbl, pi ).ATLAS, "\n" ); > else > Print( PermCharInfoRelative( tbl, tbl2, pi ).ATLAS, "\n" ); > fi; > elif IsEmpty( Intersection( pi, candidates ) ) then > Print( "G = ", Identifier( tbl2 ), ": no ", Identifier( s ), "\n" ); > else > Error( "problem with verify" ); > fi; > end;; </pre> <div class="p"><!----></div> Since in most cases the character tables of possible point stabilizers are contained in the <font face="helvetica">GAP</font> Character Table Library, the above function provides an easy test. Alternatively, we could compute <em>all</em> faithful possible permutation characters (not only the multiplicity-free ones) of the degree in question; if there are as many different such characters as are known to be induced from point stabilizers <em>and</em> if no other subgroups of this index exist then the characters are indeed permutation characters, and we can compare them with the multiplicity-free characters computed before. <div class="p"><!----></div> In the verification of the candidates, the following situations occur. <div class="p"><!----></div> <b>Lemma 1</b> <em><a name="situationI"> </a> Let Φ:[^G] → G be a group epimorphism, with K = ker(Φ) cyclic of order m, and let H be a subgroup of G such that m is coprime to the order of the commutator factor group of H. Assume that it is known that Φ<sup>−1</sup>(H) is a direct product of H with K. (This holds for example if H is simple and the order of the Schur multiplier of H is coprime to m.) Then the preimages under Φ of the G-conjugates of H contain one [^G]-class of subgroups that are isomorphic with H and that intersect trivially with K. <div class="p"><!----></div> </em> <b>Lemma 2</b> <em><a name="situationII"> </a> Let Φ:[^G] → G be a group epimorphism, with K = ker(Φ) of order 3, such that the derived subgroup G<sup>′</sup> of G has index 2 in G and such that K is not central in [^G]. (So Φ<sup>−1</sup>(G<sup>′</sup>) is the centralizer of K in [^G].) Consider a subgroup H of G with a subgroup H<sub>0</sub> = H ∩G<sup>′</sup> of index 2 in H, and assume that the preimage Φ<sup>−1</sup>(H<sub>0</sub>) is a direct product of H<sub>0</sub> with K. (This holds for example if H<sub>0</sub> is simple and the order of the Schur multiplier of H<sub>0</sub> is coprime to 3.) Then each complement of K in Φ<sup>−1</sup>(H<sub>0</sub>) extends in Φ<sup>−1</sup>(H) to exactly three complements of K that are isomorphic with H and conjugate in Φ<sup>−1</sup>(H). <div class="p"><!----></div> </em> <b>Lemma 3</b> <em><a name="situationIII"> </a> Let Φ:[^G] → G be a group epimorphism, with K = ker(Φ) of order 2. Consider a subgroup H of G, with derived subgroup H<sup>′</sup> of index 2 in H and such that Φ<sup>−1</sup>(H<sup>′</sup>) is a direct product K ×H<sup>′</sup>. <ul> <br />(i) Suppose that there is an element h ∈ H \H<sup>′</sup> such that the squares of the preimages of h in [^G] lie in the unique subgroup of index 2 in Φ<sup>−1</sup>(H<sup>′</sup>). (This holds for example if the preimages of h are involutions.) Then Φ<sup>−1</sup>(H) has the type K ×H. <br />(ii) If Φ<sup>−1</sup>(H) has the type K ×H then this group contains exactly two subgroups that are isomorphic with H. If H is a maximal subgroup of G then these two subgroups are not conjugate in [^G]. <br />(iii) Suppose that case (ii) applies and that there is h ∈ H \H<sup>′</sup> whose two preimages under Φ are not conjugate in [^G] and such that each of the two subgroups of the type H in Φ<sup>−1</sup>(H) contains elements in only one conjugacy class of [^G] that contain the preimages of h. Then the two subgroups of the type H induce different permutation characters of [^G], in particular exactly two conjugacy classes of subgroups of the type H in [^G] arise from the conjugates of H in G.</ul> <div class="p"><!----></div> </em>With character theoretic methods, we can check a weaker form of Lemma <a href="#situationIII">2.3</a> (i). Namely, the conditions are clearly satisfied if there is a conjugacy class C of elements in H that is not contained in H<sup>′</sup> and such that the class of [^G] that contains the squares of the preimages of C is <em>not</em> contained in the images of the classes of 2 ×H<sup>′</sup> that lie outside H<sup>′</sup>. <div class="p"><!----></div> The function <tt>CheckConditionsForLemma3</tt> tests this, and prints a message if Lemma <a href="#situationIII">2.3</a> (i) applies because of this situation. More precisely, the arguments are (in this order) the character tables of H<sup>′</sup>, H, G, [^G], and one of the strings <tt>"all"</tt>, <tt>"extending"</tt>; the last argument expresses that either all embeddings of H into G are considered or only those which do not lie inside the derived subgroup of G. <div class="p"><!----></div> The function <em>assumes</em> that <tt>s0</tt> is the character table of the derived subgroup of the group of <tt>s</tt>, and that H<sup>′</sup> lifts to a direct product in [^G]. <div class="p"><!----></div> <pre> gap> CheckConditionsForLemma3:= function( s0, s, fact, tbl, admissible ) > local s0fuss, poss, der, sfusfact, outerins, outerinfact, preim, > squares, dp, dpfustbl, s0indp, other, goodclasses; > > if Size( s ) <> 2 * Size( s0 ) then > Error( "<s> must be twice as large as <s0>" ); > fi; > > s0fuss:= GetFusionMap( s0, s ); > if s0fuss = fail then > poss:= Set( List( PossiblePermutationCharacters( s0, s ), > pi -> Filtered( [ 1 .. Length( pi ) ], > i -> pi[i] <> 0 ) ) ); > if Length( poss ) = 1 then > s0fuss:= poss[1]; > else > Error( "classes of <s0> in <s> not determined" ); > fi; > fi; > sfusfact:= PossibleClassFusions( s, fact ); > if admissible = "extending" then > der:= ClassPositionsOfDerivedSubgroup( fact ); > sfusfact:= Filtered( sfusfact, map -> not IsSubset( der, map ) ); > fi; > outerins:= Difference( [ 1 .. NrConjugacyClasses( s ) ], s0fuss ); > outerinfact:= Set( List( sfusfact, map -> Set( map{ outerins } ) ) ); > if Length( outerinfact ) <> 1 then > Error( "classes of `", s, "' inside `", fact, "' not determined" ); > fi; > > preim:= Flat( InverseMap( GetFusionMap( tbl, fact ) ){ outerinfact[1] } ); > squares:= Set( PowerMap( tbl, 2 ){ preim } ); > dp:= s0 * CharacterTable( "Cyclic", 2 ); > dpfustbl:= PossibleClassFusions( dp, tbl ); > s0indp:= GetFusionMap( s0, dp ); > other:= Difference( [ 1 .. NrConjugacyClasses( dp ) ], s0indp ); > goodclasses:= List( dpfustbl, map -> Intersection( squares, > Difference( map{ s0indp }, map{ other } ) ) ); > if not IsEmpty( Intersection( goodclasses ) ) then > Print( Identifier( tbl ), ": ", Identifier( s ), > " lifts to a direct product,\n", > "proved by squares in ", Intersection( goodclasses ), ".\n" ); > elif ForAll( goodclasses, IsEmpty ) then > Print( Identifier( tbl ), ": ", Identifier( s ), > " lifts to a nonsplit extension.\n" ); > else > Print( "sorry, no proof of the splitting!\n" ); > fi; > end;; </pre> <div class="p"><!----></div> Lemma <a href="#situationIII">2.3</a> (iii) can be utilized as follows. We assume the situation of Lemma <a href="#situationIII">2.3</a>, so Φ<sup>−1</sup>(H) is a direct product 〈z 〉×H, where z is an involution. The derived subgroup of Φ<sup>−1</sup>(H) is H<sub>0</sub> ≅ H<sup>′</sup>, and Φ<sup>−1</sup>(H) contains two subgroups H<sub>1</sub>, H<sub>2</sub> which are isomorphic with H, and such that H<sub>2</sub> = H<sub>0</sub> ∪{ h z; h ∈ H<sub>1</sub> \H<sub>0</sub> }. If the embedding of H<sub>1</sub>, say, into [^G] has the properties that an element outside H<sub>0</sub> is mapped into a class C of [^G] that is different from z C and such that no element of H<sub>1</sub> lies in z C then z C contains elements of H<sub>2</sub> but C does not. In particular, the permutation characters of the two actions of [^G] on the cosets of H<sub>1</sub> and H<sub>2</sub>, respectively, are necessarily different. <div class="p"><!----></div> We check this with the following function. Its arguments are one class fusion from the character table of H<sub>1</sub> to that of [^G], the factor fusion from the character table of [^G] to that of G, and the list of positions of the classes of H<sub>0</sub> in the character table of H<sub>1</sub>. The return value is <tt>true</tt> if there are two different permutation characters, and <tt>false</tt> if this cannot be proved using the criterion. <div class="p"><!----></div> <pre> gap> NecessarilyDifferentPermChars:= function( fusion, factfus, inner ) > local outer, inv; > > outer:= Difference( [ 1 .. Length( fusion ) ], inner ); > fusion:= fusion{ outer }; > inv:= Filtered( InverseMap( factfus ), IsList ); > return ForAny( inv, pair -> Length( Intersection( pair, fusion ) ) = 1 ); > end;; </pre> <div class="p"><!----></div> The following observation is used to determine the number of conjugacy classes of certain subgroups. <div class="p"><!----></div> <b>Lemma 4</b> <em><a name="conjugacy"> </a> Let G be a group with [G:G<sup>′</sup>] = 2, and Z ⊆ Z(G) < G<sup>′</sup> with |Z| = 2. Consider a maximal subgroup M of G with Z < M and M ⊄ eq G<sup>′</sup>, and a subgroup H < M with [M:H] = 4 such that U = H ∩G<sup>′</sup> is normal in M, U ≠ H holds, and Z ⊄ eq H. Let N = Z H. Then the three subgroups of index two in N that lie above U are Z U, H, and a group [H\tilde], say. If M/U is a dihedral group of order eight then the groups H and [H\tilde] are conjugate in M, and M/U is a dihedral group of order eight if and only if M \H contains both elements whose squares lie in U and elements whose squares do not lie in U. <div class="p"><!----></div> </em> <center> <img src="multfre21.png" alt="multfre21.png" /> </center> <div class="p"><!----></div> We want to detect that M/U is a dihedral group by character theoretic means but <em>without</em> using the character table of M. A sufficient (but not necessary) condition is that the set D = { g ∈ G | 1<sub>M</sub><sup>G</sup> ≠ 0, 1<sub>N</sub><sup>G</sup>(g) = 0 } is nonempty and that there are elements g<sub>1</sub>, g<sub>2</sub> ∈ D with the properties 1<sub>U</sub><sup>G</sup>(g<sub>1</sub><sup>2</sup>) = 0 and |g<sub>2</sub>| = 2. <div class="p"><!----></div> The following function takes the character table of G and the three permutation characters 1<sub>U</sub><sup>G</sup>, 1<sub>M</sub><sup>G</sup>, 1<sub>N</sub><sup>G</sup>, and returns a list of length two, the i-th entry being the list of class positions of elements that can serve as g<sub>i</sub>. So M/U is proved to be a dihedral group if both entries are nonempty. <div class="p"><!----></div> <pre> gap> ProofOfD8Factor:= function( tblG, piU, piM, piN ) > local D, map, D1, D2; > > D:= Filtered( [ 1 .. Length( piU ) ], i -> piM[i] <> 0 and piN[i] = 0 ); > map:= PowerMap( tblG, 2 ); > D1:= Filtered( D, i -> piU[ map[i] ] = 0 ); > D2:= Filtered( D, i -> OrdersClassRepresentatives( tblG )[i] = 2 ); > return [ D1, D2 ]; > end;; </pre> <div class="p"><!----></div> <div class="p"><!----></div> <h3><a name="tth_sEc2.3"> 2.3</a> Isoclinic Groups</h3> <div class="p"><!----></div> For dealing with the character tables of groups of the type 2.G.2 that are isoclinic to those whose tables are printed in the A<font size="-2">TLAS</font> ([<a href="#CCN85 it is necessary to store explicitly the factor fusion from 2.G.2 onto G.2 and the subgroup fusion from 2.G into 2.G.2, in order to make the above functions work. Note that these maps coincide for the two isoclinism types. <div class="p"><!----></div> <pre> gap> IsoclinicTable:= function( tbl, tbl2, facttbl ) > local subfus, factfus; > > subfus:= GetFusionMap( tbl, tbl2 ); > factfus:= GetFusionMap( tbl2, facttbl ); > tbl2:= CharacterTableIsoclinic( tbl2 ); > StoreFusion( tbl, subfus, tbl2 ); > StoreFusion( tbl2, factfus, facttbl ); > return tbl2; > end;; </pre> <div class="p"><!----></div> <h3><a name="tth_sEc2.4"> 2.4</a> Tests for <font face="helvetica">GAP</font></h3> <div class="p"><!----></div> With the following function, we check whether the characters computed here coincide with the characters stored in the data file. <div class="p"><!----></div> <pre> gap> CompareWithDatabase:= function( name, chars ) > local info; > > info:= MultFreeEndoRingCharacterTables( name ); > info:= List( info, x -> x.character );; > if SortedList( info ) <> SortedList( Concatenation( chars ) ) then > Error( "contradiction 1 for ", name ); > fi; > end;; </pre> <div class="p"><!----></div> If the character tables of all maximal subgroups of G are known then we could use alternatively the same method (and in fact the same <font face="helvetica">GAP</fon functions) as in the classification in [<a href="#BL96" name="CITEBL96">BL96</a>]. This is shown in the following sections where applicable, using the following function. (The function <tt>PossiblePermutationCharactersWithBoundedMultiplicity</tt> is defined in the file <tt>tst/multfree.g</tt> of the <font face="helvetica">GAP</font> Character Table Library [<a href="#CTblLib" name="CITECTbl note that it returns not only faithful characters.) <div class="p"><!----></div> <pre> gap> CompareWithCandidatesByMaxes:= function( name, faith ) > local tbl, poss; > > tbl:= CharacterTable( name ); > if not HasMaxes( tbl ) then > Error( "no maxes stored for ", name ); > fi; > poss:= PossiblePermutationCharactersWithBoundedMultiplicity( tbl, 1 ); > poss:= List( poss.permcand, l -> Filtered( l, > pi -> ClassPositionsOfKernel( pi ) = [ 1 ] ) ); > if SortedList( Concatenation( poss ) ) > <> SortedList( Concatenation( faith ) ) then > Error( "contradiction 2 for ", name ); > fi; > end;; </pre> <div class="p"><!----></div> <h2><a name="tth_sEc3"> 3</a> The Groups</h2> <div class="p"><!----></div> In the following, we use A<font size="-2">TLAS</font> notation (see [<a href="#CCN85" name="CITECCN85">CCN<sup>+</s In particular, 2 ×G and G ×2 denote the direct product of the group G with a cyclic group of order 2, and G.2 and 2.G denote an upward and downward extension, respectively, of G by a cyclic group of order 2, such that these groups are <em>not</em> direct products. <div class="p"><!----></div> For groups of the structure 2.G.2 where the character table of G is contained in the A<font size="-2">TLAS</font>, we use the name 2.G.2 for the isoclinism type whose character table is printed in the A<font size="-2">TLAS</font>, and (2.G.2)<sup>∗</sup> for the other isoclinism type. <div class="p"><!----></div> Most of the computations that are shown in the following use only information from the <font face="helvetica">GAP</font> Character Table Library. The (few) explicit computations with groups are collected in Section <a href="#explicit">4</a>. <div class="p"><!----></div> <h3><a name="tth_sEc3.1"> 3.1</a> G = 2.M<sub>12</sub></h3> <div class="p"><!----></div> The group 2.M<sub>12</sub> has ten faithful multiplicity-free permutation actions, with point stabilizers of the types M<sub>11</sub> (twice), A<sub>6</sub>.2<sub>1</sub> (twice), 3<sup>2</sup>.2.S<sub>4</sub> (four classes), and 3<sup>2</sup>:2.A<su <div class="p"><!----></div> <pre> gap> tbl:= CharacterTable( "2.M12" );; gap> faith:= FaithfulCandidates( tbl, "M12" );; 1: subgroup $M_{11}$, degree 24 (1 cand.) 2: subgroup $M_{11}$, degree 24 (1 cand.) 5: subgroup $A_6.2_1 \leq A_6.2^2$, degree 264 (1 cand.) 8: subgroup $A_6.2_1 \leq A_6.2^2$, degree 264 (1 cand.) 11: subgroup $3^2.2.S_4$, degree 440 (2 cand.) 12: subgroup $3^2:2.A_4 \leq 3^2.2.S_4$, degree 880 (1 cand.) 13: subgroup $3^2.2.S_4$, degree 440 (2 cand.) 14: subgroup $3^2:2.A_4 \leq 3^2.2.S_4$, degree 880 (1 cand.) </pre> <div class="p"><!----></div> There are two classes of M<sub>11</sub> subgroups in M<sub>12</sub> as well as in 2.M<sub>12</sub>, so we apply Lemma <a href="#situationI">2.1</a>. <div class="p"><!----></div> <pre> gap> VerifyCandidates( CharacterTable( "M11" ), tbl, 0, > Concatenation( faith[1], faith[2] ), "all" ); G = 2.M12: point stabilizer M11, ranks [ 3, 3 ] [ "1a+11a+12a", "1a+11b+12a" ] </pre> <div class="p"><!----></div> According to the list of maximal subgroups of 2.M<sub>12</sub>, any A<sub>6</sub>.2<sup>2</sup> subgroup in M<sub>12</sub> lifts to a group of the structure A<sub>6</sub>.D<sub>8</sub> in M<sub>12</sub>, which contains two conjugate subgroups of the type A<sub>6</sub>.2<sub>1</sub>; so we get two classes of such subgroups, with the same permutation character. <div class="p"><!----></div> <pre> gap> Maxes( tbl ); [ "2xM11", "2.M12M2", "A6.D8", "2.M12M4", "2.L2(11)", "2x3^2.2.S4", "2.M12M7", "2.M12M8", "2.M12M9", "2.M12M10", "2.A4xS3" ] gap> faith[5] = faith[8]; true gap> VerifyCandidates( CharacterTable( "A6.2_1" ), tbl, 0, faith[5], "all" ); G = 2.M12: point stabilizer A6.2_1, ranks [ 7 ] [ "1a+11ab+12a+54a+55a+120b" ] </pre> <div class="p"><!----></div> The 3<sup>2</sup>.2.S<sub>4</sub> type subgroups of M<sub>12</sub> lift to direct products with the centre of 2.M<sub>12</sub>, each such group contains two subgroups of the type 3<sup>2</sup>.2.S<sub>4</sub> which induce different permutation characters, for example because the involutions in 3<sup>2</sup>.2.S<sub>4</sub> \3<sup>2</sup>.2.A<sub>4</sub> lie in the two preimages of the class <tt>2B</tt> of M<sub>12</sub>. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "3^2.2.S4" );; gap> derpos:= ClassPositionsOfDerivedSubgroup( s );; gap> facttbl:= CharacterTable( "M12" );; gap> factfus:= GetFusionMap( tbl, facttbl );; gap> ForAll( PossibleClassFusions( s, tbl ), > map -> NecessarilyDifferentPermChars( map, factfus, derpos ) ); true gap> VerifyCandidates( s, tbl, 0, Concatenation( faith[11], faith[13] ), "all" ); G = 2.M12: point stabilizer 3^2.2.S4, ranks [ 7, 7, 9, 9 ] [ "1a+11a+54a+55a+99a+110ab", "1a+11b+54a+55a+99a+110ab", "1a+11a+12a+44ab+54a+55a+99a+120b", "1a+11b+12a+44ab+54a+55a+99a+120b" ] </pre> <div class="p"><!----></div> Each 3<sup>2</sup>.2.S<sub>4</sub> type group contains a unique subgroup of the type 3<sup>2</sup>.2.A<sub>4</sub>, we get two classes of such subgroups, with different permutation characters because already the corresponding characters for M<sub>12</sub> are different; we verify the candidates by inducing the degree two permutation characters of the 3<sup>2</sup>.2.S<sub>4</sub> type groups to 2.M<sub>12</sub>. <div class="p"><!----></div> <pre> gap> fus:= PossibleClassFusions( s, tbl );; gap> deg2:= PermChars( s, 2 ); [ Character( CharacterTable( "3^2.2.S4" ), [ 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0 ] ) ] gap> pi:= Set( List( fus, map -> Induced( s, tbl, deg2, map )[1] ) );; gap> pi = SortedList( Concatenation( faith[12], faith[14] ) ); true gap> PermCharInfo( tbl, pi ).ATLAS; [ "1a+11a+12a+44ab+45a+54a+55ac+99a+110ab+120ab", "1a+11b+12a+44ab+45a+54a+55ab+99a+110ab+120ab" ] gap> CompareWithDatabase( "2.M12", faith ); gap> CompareWithCandidatesByMaxes( "2.M12", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.2"> 3.2</a> G = 2.M<sub>12</sub>.2</h3> <div class="p"><!----></div> The group 2.M<sub>12</sub>.2 that is printed in the A<font size="-2">TLAS</font> has three faithful multiplicity-free permutation actions, with point stabilizers of the types M<sub>11</sub> and L<sub>2</sub>(11).2 (twice), respectively. <div class="p"><!----></div> <pre> gap> tbl2:= CharacterTable( "2.M12.2" );; gap> faith:= FaithfulCandidates( tbl2, "M12.2" );; 1: subgroup $M_{11}$, degree 48 (1 cand.) 2: subgroup $L_2(11).2$, degree 288 (2 cand.) </pre> <div class="p"><!----></div> The two classes of subgroups of the type M<sub>11</sub> in 2.M<sub>12</sub> are fused in 2.M<sub>12</sub>.2, so we get one class of these subgroups. <div class="p"><!----></div> <pre> gap> VerifyCandidates( CharacterTable( "M11" ), tbl, tbl2, faith[1], "all" ); G = 2.M12.2: point stabilizer M11, ranks [ 5 ] [ "1a^{\\pm}+11ab+12a^{\\pm}" ] </pre> <div class="p"><!----></div> The outer involutions in the maximal subgroups of the type L<sub>2</sub>(11).2 in M<sub>12</sub>.2 lift to involutions in 2.M<sub>12</sub>.2; moreover, those subgroups of the type L<sub>2</sub>(11).2 that are novelties (so the intersection with M<sub>12</sub> lies in M<sub>11</sub> type subgroups) contain <tt>2B</tt> elements, which lift to involutions in 2.M<sub>12</sub>.2, so the L<sub>2</sub>(11) subgroup lifts to a group of the type 2 ×L<sub>2</sub>(11), and Lemma <a href="#situationIII">2.3</a> (ii) yields two classes of subgroups. The permutation characters are different, for example because each of the two candidates contains elements in one of the two preimages of the class <tt>2B</tt>. <div class="p"><!----></div> (The function <tt>CheckConditionsForLemma3</tt> fails here, because of the two classes of maximal subgroups L<sub>2</sub>(11).2 in M<sub>12</sub>.2. One of them contains <tt>2A</tt> elements, the other contains <tt>2B</tt> elements. Only the latter type of subgroups, whose intersection with M<sub>12</sub> is not maximal in M<sub>12</sub>, lifts to subgroups of 2.M<sub>12</sub>.2 that contain L<sub>2</sub>(11).2 subgroups.) <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "L2(11).2" );; gap> derpos:= ClassPositionsOfDerivedSubgroup( s );; gap> facttbl:= CharacterTable( "M12.2" );; gap> factfus:= GetFusionMap( tbl2, facttbl );; gap> ForAll( PossibleClassFusions( s, tbl2 ), > map -> NecessarilyDifferentPermChars( map, factfus, derpos ) ); true gap> VerifyCandidates( s, tbl, tbl2, faith[2], "all" ); G = 2.M12.2: point stabilizer L2(11).2, ranks [ 7, 7 ] [ "1a^++11ab+12a^{\\pm}+55a^++66a^++120b^-", "1a^++11ab+12a^{\\pm}+55a^++66a^++120b^+" ] gap> CompareWithDatabase( "2.M12.2", faith ); </pre> <div class="p"><!----></div> The group (2.M<sub>12</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed in the A<font size="-2">TLAS</font> has one faithful multiplicity-free permutation action, with point stabilizer of the type M<sub>11</sub>; as this subgroup lies inside 2.M<sub>12</sub>, its existence is clear, and the permutation character in both groups of the type 2.M<sub>12</sub>.2 is the same. <div class="p"><!----></div> <pre> gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );; gap> faith:= FaithfulCandidates( tbl2, "M12.2" );; 1: subgroup $M_{11}$, degree 48 (1 cand.) gap> CompareWithDatabase( "Isoclinic(2.M12.2)", faith ); </pre> <div class="p"><!----></div> Note that in (2.M<sub>12</sub>.2)<sup>∗</sup>, the subgroup of the type (2 ×L<sub>2</sub>(11)).2 is a nonsplit extension, so the unique index 2 subgroup in this group contains the centre of 2.M<sub>12</sub>.2, in particular there is no subgroup of the type L<sub>2</sub>(11).2. <div class="p"><!----></div> <pre> gap> PossibleClassFusions( CharacterTable( "L2(11).2" ), tbl2 ); [ ] </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.3"> 3.3</a> G = 2.M<sub>22</sub></h3><a name="libtbl"> </a> <div class="p"><!----></div> The group 2.M<sub>22</sub> has four faithful multiplicity-free permutation actions, with point stabilizers of the types 2<sup>4</sup>:A<sub>5</sub>, A<sub>7</sub> (twice), and 2<sup>3</sup>:L<sub>3</sub>(2), by Lemma <a href="#situationI">2.1</a>. <div class="p"><!----></div> <pre> gap> tbl:= CharacterTable( "2.M22" );; gap> faith:= FaithfulCandidates( tbl, "M22" );; 3: subgroup $2^4:A_5 \leq 2^4:A_6$, degree 924 (1 cand.) 4: subgroup $A_7$, degree 352 (1 cand.) 5: subgroup $A_7$, degree 352 (1 cand.) 7: subgroup $2^3:L_3(2)$, degree 660 (1 cand.) </pre> <div class="p"><!----></div> Note that one class of subgroups of the type 2<sup>4</sup>:A<sub>5</sub> in the maximal subgroup of the type 2<sup>4</sup>:A<sub>6</sub> as well as the A<sub>7</sub> and 2<sup>3</sup>:L<sub>3</sub>(2) subgrou lift to direct products in 2.M<sub>22</sub>. A proof for 2<sup>4</sup>:A<sub>5</sub> using explicit computations with the group can be found in Section <a href="#explicit1">4.1</a>. <div class="p"><!----></div> <pre> gap> Maxes( tbl ); [ "2.L3(4)", "2.M22M2", "2xA7", "2xA7", "2.M22M5", "2x2^3:L3(2)", "(2xA6).2_3", "2xL2(11)" ] gap> s:= CharacterTable( "P1/G1/L1/V1/ext2" );; gap> VerifyCandidates( s, tbl, 0, faith[3], "all" ); G = 2.M22: point stabilizer P1/G1/L1/V1/ext2, ranks [ 8 ] [ "1a+21a+55a+126ab+154a+210b+231a" ] gap> faith[4] = faith[5]; true gap> VerifyCandidates( CharacterTable( "A7" ), tbl, 0, faith[4], "all" ); G = 2.M22: point stabilizer A7, ranks [ 5 ] [ "1a+21a+56a+120a+154a" ] gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[7], "all" ); G = 2.M22: point stabilizer 2^3:sl(3,2), ranks [ 7 ] [ "1a+21a+55a+99a+120a+154a+210b" ] gap> CompareWithDatabase( "2.M22", faith ); gap> CompareWithCandidatesByMaxes( "2.M22", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.4"> 3.4</a> G = 2.M<sub>22</sub>.2</h3><a name="2.M22.2"> </a> <div class="p"><!----></div> The group 2.M<sub>22</sub>.2 that is printed in the A<font size="-2">TLAS</font> has eight faithful multiplicity-free permutation actions, with point stabilizers of the types 2<sup>4</sup>:S<sub>5</sub> (twice), A<sub>7</sub>, 2<sup>3</sup>:L<sub>3</sub>(2) ×2 (twice), 2<sup>3</sup>:L<sub>3</sub>(2), and L<sub>2</sub>(11).2 (twice). <div class="p"><!----></div> <pre> gap> tbl2:= CharacterTable( "2.M22.2" );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; 6: subgroup $2^4:S_5 \leq 2^4:S_6$, degree 924 (2 cand.) 7: subgroup $A_7$, degree 704 (1 cand.) 11: subgroup $2^3:L_3(2) \times 2$, degree 660 (2 cand.) 12: subgroup $2^3:L_3(2) \leq 2^3:L_3(2) \times 2$, degree 1320 (2 cand.) 16: subgroup $L_2(11).2$, degree 1344 (2 cand.) </pre> <div class="p"><!----></div> The character table of the 2<sup>4</sup>:S<sub>5</sub> type subgroup is contained in the <font face="h Character Table Library, with identifier <tt>w(d5)</tt> (which denotes the Weyl group of the type D<sub>5</sub>, cf. Section <a href="#explicit2">4.2</a>). <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "w(d5)" );; gap> derpos:= ClassPositionsOfDerivedSubgroup( s );; gap> facttbl:= CharacterTable( "M22.2" );; gap> factfus:= GetFusionMap( tbl2, facttbl );; gap> ForAll( PossibleClassFusions( s, tbl2 ), > map -> NecessarilyDifferentPermChars( map, factfus, derpos ) ); true gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" ); G = 2.M22.2: point stabilizer w(d5), ranks [ 7, 7 ] [ "1a^++21a^++55a^++126ab+154a^++210b^-+231a^-", "1a^++21a^++55a^++126ab+154a^++210b^++231a^-" ] </pre> <div class="p"><!----></div> The two classes of the type A<sub>7</sub> subgroups in 2.M<sub>22</sub> are fused in 2.M<sub>22</sub>.2. <div class="p"><!----></div> <pre> gap> VerifyCandidates( CharacterTable( "A7" ), tbl, tbl2, faith[7], "all" ); G = 2.M22.2: point stabilizer A7, ranks [ 10 ] [ "1a^{\\pm}+21a^{\\pm}+56a^{\\pm}+120a^{\\pm}+154a^{\\pm}" ] </pre> <div class="p"><!----></div> The preimages of the 2<sup>3</sup>:L<sub>3</sub>(2) ×2 type subgroups of M<sub>22</sub>.2 in 2.M<sub>22</sub>.2 are direct products, by the discussion of 2.M<sub>22</sub> and Lemma <a href="#situationIII">2.3</a> (i). So Lemma <a href="#situationIII">2.3</a> (iii) yields two classes, with different permutation characters. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2x2^3:L3(2)" );; gap> s0:= CharacterTable( "2^3:sl(3,2)" );; gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" ); 2.M22.2: 2x2^3:L3(2) lifts to a direct product, proved by squares in [ 1, 5, 14, 16 ]. gap> derpos:= ClassPositionsOfDerivedSubgroup( s );; gap> ForAll( PossibleClassFusions( s, tbl2 ), > map -> NecessarilyDifferentPermChars( map, factfus, derpos ) ); true gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" ); G = 2.M22.2: point stabilizer 2x2^3:L3(2), ranks [ 7, 7 ] [ "1a^++21a^++55a^++99a^++120a^-+154a^++210b^-", "1a^++21a^++55a^++99a^++120a^++154a^++210b^+" ] </pre> <div class="p"><!----></div> There is one class of subgroups of the type 2<sup>3</sup>:L<sub>3</sub>(2) in 2.M<sub>22</sub>. One of the two candidates of degree 1 320 is excluded because it does not arise from a possible class fusion. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "M22M6" );; gap> pi1320:= PossiblePermutationCharacters( s, tbl2 );; gap> Length( pi1320 ); 1 gap> IsSubset( faith[12], pi1320 ); true gap> faith[12]:= pi1320;; gap> VerifyCandidates( s, tbl, tbl2, faith[12], "all" ); G = 2.M22.2: point stabilizer 2^3:sl(3,2), ranks [ 14 ] [ "1a^{\\pm}+21a^{\\pm}+55a^{\\pm}+99a^{\\pm}+120a^{\\pm}+154a^{\\pm}+210b^{\\\ pm}" ] </pre> <div class="p"><!----></div> By Lemma <a href="#situationIII">2.3</a> (i), the preimages of the L<sub>2</sub>(11).2 type subgroups of M<sub>22</sub>.2 in 2.M<sub>22</sub>.2 are direct products, so Lemma <a href="#situationIII">2.3</a> (iii) yields two classes, with different permutation characters. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "L2(11).2" );; gap> s0:= CharacterTable( "L2(11)" );; gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" ); 2.M22.2: L2(11).2 lifts to a direct product, proved by squares in [ 1, 4, 10, 13 ]. gap> derpos:= ClassPositionsOfDerivedSubgroup( s );; gap> ForAll( PossibleClassFusions( s, tbl2 ), > map -> NecessarilyDifferentPermChars( map, factfus, derpos ) ); true gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" ); G = 2.M22.2: point stabilizer L2(11).2, ranks [ 10, 10 ] [ "1a^++21a^-+55a^++56a^{\\pm}+120a^-+154a^++210a^-+231a^-+440a^+", "1a^++21a^-+55a^++56a^{\\pm}+120a^++154a^++210a^-+231a^-+440a^-" ] gap> CompareWithDatabase( "2.M22.2", faith ); </pre> <div class="p"><!----></div> The group (2.M<sub>22</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed in the A<font size="-2">TLAS</font> has two faithful multiplicity-free permutation actions, with point stabilizers of the types A<sub>7</sub> and 2<sup>3</sup>:L<sub>3</sub>(2). <div class="p"><!----></div> <pre> gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; 7: subgroup $A_7$, degree 704 (1 cand.) 12: subgroup $2^3:L_3(2) \leq 2^3:L_3(2) \times 2$, degree 1320 (2 cand.) gap> faith[12]:= Filtered( faith[12], chi -> chi in pi1320 );; gap> CompareWithDatabase( "Isoclinic(2.M22.2)", faith ); </pre> <div class="p"><!----></div> The two classes of subgroups lie inside 2.M<sub>22</sub>, so their existence has been discussed already above. <div class="p"><!----></div> <h3><a name="tth_sEc3.5"> 3.5</a> G = 3.M<sub>22</sub></h3> <div class="p"><!----></div> The group 3.M<sub>22</sub> has four faithful multiplicity-free permutation actions, with point stabilizers of the types 2<sup>4</sup>:A<sub>5</sub>, 2<sup>4</sup>:S<sub>5</sub>, 2<sup>3</sup>: and L<sub>2</sub>(11). <div class="p"><!----></div> <pre> gap> tbl:= CharacterTable( "3.M22" );; gap> faith:= FaithfulCandidates( tbl, "M22" );; 3: subgroup $2^4:A_5 \leq 2^4:A_6$, degree 1386 (1 cand.) 6: subgroup $2^4:S_5$, degree 693 (1 cand.) 7: subgroup $2^3:L_3(2)$, degree 990 (1 cand.) 9: subgroup $L_2(11)$, degree 2016 (1 cand.) </pre> <div class="p"><!----></div> The existence of one class of each of these types follows from Lemma <a href="#situationI">2.1</a>. <div class="p"><!----></div> <pre> gap> VerifyCandidates( CharacterTable( "P1/G1/L1/V1/ext2" ), tbl, 0, faith[3], "all" ); G = 3.M22: point stabilizer P1/G1/L1/V1/ext2, ranks [ 13 ] [ "1a+21abc+55a+105abcd+154a+231abc" ] gap> VerifyCandidates( CharacterTable( "M22M5" ), tbl, 0, faith[6], "all" ); G = 3.M22: point stabilizer 2^4:s5, ranks [ 10 ] [ "1a+21abc+55a+105abcd+154a" ] gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[7], "all" ); G = 3.M22: point stabilizer 2^3:sl(3,2), ranks [ 13 ] [ "1a+21abc+55a+99abc+105abcd+154a" ] gap> VerifyCandidates( CharacterTable( "M22M8" ), tbl, 0, faith[9], "all" ); G = 3.M22: point stabilizer L2(11), ranks [ 16 ] [ "1a+21abc+55a+105abcd+154a+210abc+231abc" ] gap> CompareWithDatabase( "3.M22", faith ); gap> CompareWithCandidatesByMaxes( "3.M22", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.6"> 3.6</a> G = 3.M<sub>22</sub>.2</h3> <div class="p"><!----></div> The group 3.M<sub>22</sub>.2 has five faithful multiplicity-free permutation actions, with point stabilizers of the types 2<sup>4</sup>:S<sub>5</sub>, 2<sup>5</sup>:S<sub>5</sub>, 2<sup>4</sup>:(A<sub>5</sub> ×2), 2<sup>3</sup>:L<sub>3</sub>(2) ×2, and L<sub>2</sub>(11).2. <div class="p"><!----></div> <pre> gap> tbl2:= CharacterTable( "3.M22.2" );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; 6: subgroup $2^4:S_5 \leq 2^4:S_6$, degree 1386 (1 cand.) 8: subgroup $2^5:S_5$, degree 693 (1 cand.) 10: subgroup $2^4:(A_5 \times 2) \leq 2^5:S_5$, degree 1386 (1 cand.) 11: subgroup $2^3:L_3(2) \times 2$, degree 990 (1 cand.) 16: subgroup $L_2(11).2$, degree 2016 (1 cand.) </pre> <div class="p"><!----></div> Subgroups of these types exist by Lemma <a href="#situationII">2.2</a>. The verification is straightforward in all cases except that of 2<sup>4</sup>:(A<sub>5</sub> ×2). <div class="p"><!----></div> <pre> gap> VerifyCandidates( CharacterTable( "w(d5)" ), tbl, tbl2, faith[6], "all" ); G = 3.M22.2: point stabilizer w(d5), ranks [ 9 ] [ "1a^++21a^+bc+55a^++105abcd+154a^++231a^-bc" ] gap> VerifyCandidates( CharacterTable( "M22.2M4" ), tbl, tbl2, faith[8], "all" ); G = 3.M22.2: point stabilizer M22.2M4, ranks [ 7 ] [ "1a^++21a^+bc+55a^++105abcd+154a^+" ] gap> VerifyCandidates( CharacterTable( "2x2^3:L3(2)" ), tbl, tbl2, faith[11], "all" ); G = 3.M22.2: point stabilizer 2x2^3:L3(2), ranks [ 9 ] [ "1a^++21a^+bc+55a^++99a^+bc+105abcd+154a^+" ] gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" ); G = 3.M22.2: point stabilizer L2(11).2, ranks [ 11 ] [ "1a^++21a^-bc+55a^++105abcd+154a^++210a^-bc+231a^-bc" ] </pre> <div class="p"><!----></div> In the remaining case, we note that the 2<sup>4</sup>:(A<sub>5</sub> ×2) type subgroup has index 2 in the maximal subgroup of the type 2<sup>5</sup>:S<sub>5</sub>, whose character table is available via the identifier <tt>M22.2M4</tt>. It is sufficient to show that exactly one of the three index 2 subgroups in this group induces a multiplicity-free permutation character of 3.M<sub>22</sub>.2, and this can be done by inducing the degree 2 permutation characters of 2<sup>5</sup>:S<sub>5</sub> to 3.M<sub>22</sub>.2. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "M22.2M4" );; gap> lin:= LinearCharacters( s ); [ Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) , Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 ] ), Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1 ] ), Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1 ] ) ] gap> perms:= List( lin{ [ 2 .. 4 ] }, chi -> chi + lin[1] );; gap> sfustbl2:= PossibleClassFusions( s, tbl2 );; gap> Length( sfustbl2 ); 2 gap> ind1:= Induced( s, tbl2, perms, sfustbl2[1] );; gap> ind2:= Induced( s, tbl2, perms, sfustbl2[2] );; gap> PermCharInfo( tbl2, ind1 ).ATLAS; [ "1ab+21ab+42aa+55ab+154ab+210ccdd", "1a+21ab+42a+55a+154a+210bcd+462a", "1a+21aa+42a+55a+154a+210acd+462a" ] gap> PermCharInfo( tbl2, ind2 ).ATLAS; [ "1a+21aa+42a+55a+154a+210acd+462a", "1a+21ab+42a+55a+154a+210bcd+462a", "1ab+21ab+42aa+55ab+154ab+210ccdd" ] gap> ind1[2] = ind2[2]; true gap> [ ind1[2] ] = faith[10]; true gap> CompareWithDatabase( "3.M22.2", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.7"> 3.7</a> G = 4.M<sub>22</sub> and G = 12.M<sub>22</sub></h3> <div class="p"><!----></div> The group 4.M<sub>22</sub> and hence also the group 12.M<sub>22</sub> has no faithful multiplicity-free permutation action. <div class="p"><!----></div> <pre> gap> tbl:= CharacterTable( "4.M22" );; gap> faith:= FaithfulCandidates( tbl, "2.M22" );; gap> CompareWithDatabase( "4.M22", faith ); gap> CompareWithCandidatesByMaxes( "4.M22", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.8"> 3.8</a> G = 4.M<sub>22</sub>.2 and G = 12.M<sub>22</sub>.2</h3> <div class="p"><!----></div> The two isoclinism types of groups of the type 4.M<sub>22</sub>.2 and hence also all groups of the type 12.M<sub>22</sub>.2 have no faithful multiplicity-free permutation actions. <div class="p"><!----></div> <pre> gap> tbl2:= CharacterTable( "4.M22.2" );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; gap> CompareWithDatabase( "4.M22.2", faith ); gap> CompareWithDatabase( "12.M22.2", [] ); gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; gap> CompareWithDatabase( "Isoclinic(4.M22.2)", faith ); gap> CompareWithDatabase( "Isoclinic(12.M22.2)", [] ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.9"> 3.9</a> G = 6.M<sub>22</sub></h3> <div class="p"><!----></div> The group 6.M<sub>22</sub> has two faithful multiplicity-free permutation actions, with point stabilizers of the types 2<sup>4</sup>:A<sub>5</sub> and 2<sup>3</sup>:L<sub>3</sub>(2). <div class="p"><!----></div> <pre> gap> tbl:= CharacterTable( "6.M22" );; gap> faith:= FaithfulCandidates( tbl, "3.M22" );; 1: subgroup $2^4:A_5 \rightarrow (M_{22},3)$, degree 2772 (1 cand.) 3: subgroup $2^3:L_3(2) \rightarrow (M_{22},7)$, degree 1980 (1 cand.) </pre> <div class="p"><!----></div> The existence of one class of each of these subgroups follows from the treatment of 2.M<sub>22</sub> and 3.M<sub>22</sub>. <div class="p"><!----></div> <pre> gap> VerifyCandidates( CharacterTable( "P1/G1/L1/V1/ext2" ), tbl, 0, faith[1], "all" ); G = 6.M22: point stabilizer P1/G1/L1/V1/ext2, ranks [ 22 ] [ "1a+21abc+55a+105abcd+126abcdef+154a+210bef+231abc" ] gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[3], "all" ); G = 6.M22: point stabilizer 2^3:sl(3,2), ranks [ 17 ] [ "1a+21abc+55a+99abc+105abcd+120a+154a+210b+330de" ] gap> CompareWithDatabase( "6.M22", faith ); gap> CompareWithCandidatesByMaxes( "6.M22", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.10"> 3.10</a> G = 6.M<sub>22</sub>.2</h3> <div class="p"><!----></div> The group 6.M<sub>22</sub>.2 that is printed in the A<font size="-2">TLAS</font> has six faithful multiplicity-free permutation actions, with point stabilizers of the types 2<sup>4</sup>:S<sub>5</sub> (twice), 2<sup>3</sup>:L<sub>3</sub>(2) ×2 (twice), and L<sub>2</sub>(11).2 (twice). <div class="p"><!----></div> <pre> gap> tbl2:= CharacterTable( "6.M22.2" );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; 6: subgroup $2^4:S_5 \leq 2^4:S_6$, degree 2772 (2 cand.) 11: subgroup $2^3:L_3(2) \times 2$, degree 1980 (2 cand.) 16: subgroup $L_2(11).2$, degree 4032 (2 cand.) </pre> <div class="p"><!----></div> We know that 2.M<sub>22</sub>.2 contains two classes of subgroups isomorphic with each of the required point stabilizers, so we apply Lemma <a href="#situationII">2.2</a>. <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "w(d5)" );; gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" ); G = 6.M22.2: point stabilizer w(d5), ranks [ 14, 14 ] [ "1a^++21a^+bc+55a^++105abcd+126abcdef+154a^++210b^-ef+231a^-bc", "1a^++21a^+bc+55a^++105abcd+126abcdef+154a^++210b^+ef+231a^-bc" ] </pre> <div class="p"><!----></div> (Since 6.M<sub>22</sub> contains subgroups of the type 2<sup>3</sup>:L<sub>3</sub>(2) ×2 in which we are not interested, we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt> for this case.) <div class="p"><!----></div> <pre> gap> s:= CharacterTable( "2x2^3:L3(2)" );; gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" ); G = 6.M22.2: point stabilizer 2x2^3:L3(2), ranks [ 12, 12 ] [ "1a^++21a^+bc+55a^++99a^+bc+105abcd+120a^-+154a^++210b^-+330de", "1a^++21a^+bc+55a^++99a^+bc+105abcd+120a^++154a^++210b^++330de" ] gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" ); G = 6.M22.2: point stabilizer L2(11).2, ranks [ 20, 20 ] [ "1a^++21a^-bc+55a^++56a^{\\pm}+66abcd+105abcd+120a^-bc+154a^++210a^-cdghij+2\ 31a^-bc+440a^+", "1a^++21a^-bc+55a^++56a^{\\pm}+66abcd+105abcd+120a^+bc+154a^++210a^-cdghij+2\ 31a^-bc+440a^-" ] gap> CompareWithDatabase( "6.M22.2", faith ); </pre> <div class="p"><!----></div> The group (6.M<sub>22</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed in the A<font size="-2">TLAS</font> has no faithful multiplicity-free permutation action. <div class="p"><!----></div> <pre> gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );; gap> faith:= FaithfulCandidates( tbl2, "M22.2" );; gap> CompareWithDatabase( "Isoclinic(6.M22.2)", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.11"> 3.11</a> G = 2.J<sub>2</sub></h3> <div class="p"><!----></div> The group 2.J<sub>2</sub> has one faithful multiplicity-free permutation action, with point stabilizer of the type U<sub>3</sub>(3), by Lemma <a href="#situationI">2.1</a>. <div class="p"><!----></div> <pre> gap> tbl:= CharacterTable( "2.J2" );; gap> faith:= FaithfulCandidates( tbl, "J2" );; 1: subgroup $U_3(3)$, degree 200 (1 cand.) gap> VerifyCandidates( CharacterTable( "U3(3)" ), tbl, 0, faith[1], "all" ); G = 2.J2: point stabilizer U3(3), ranks [ 5 ] [ "1a+36a+50ab+63a" ] gap> CompareWithDatabase( "2.J2", faith ); gap> CompareWithCandidatesByMaxes( "2.J2", faith ); </pre> <div class="p"><!----></div> <h3><a name="tth_sEc3.12"> 3.12</a> G = 2.J<sub>2</sub>.2</h3><a name="sect2J22"> </a> <div class="p"><!----></div> The group 2.J<sub>2</sub>.2 that is printed in the A<font size="-2">TLAS</font> has no faithful multiplicity-free permutation action. <div class="p"><!----></div> <pre> gap> tbl2:= CharacterTable( "2.J2.2" );; gap> faith:= FaithfulCandidates( tbl2, "J2.2" );; --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.43 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
|
2026-03-28