<!-- %W tutorial.xml GAP 4 package CTblLib Thomas Breuer -->
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<Chapter Label="ch:tutorial">
<Heading>Tutorial for the &GAP; Character Table Library</Heading>
This chapter gives an overview of the basic functionality
provided by the &GAP; Character Table Library.
The main concepts and interface functions are presented in
the sections <Ref Sect="sect:concepts"/> and
<Ref Sect="sect:accesstbl"/>,
Section <Ref Sect="sect:tutsectctbllib"/> shows a few small examples.
<P/>
In order to force that the output of the examples consists only of
ASCII characters,
we set the user preference <C>DisplayFunction</C>
of the <Package>AtlasRep</Package> to the value <C>"Print"</C>.
This is necessary because the &LaTeX; and HTML versions of &GAPDoc;
documents support only ASCII characters.
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<Section Label="sect:concepts">
<Heading>Concepts used in the &GAP; Character Table Library</Heading>
The main idea behind working with the &GAP; Character Table Library
is to deal with character tables of groups
but <E>without</E> having access to these groups.
This situation occurs for example if one extracts information from the
printed &ATLAS; of Finite Groups (<Cite Key="CCN85"/>).
<P/>
This restriction means first of all that we need a way to access the
character tables, see Section <Ref Sect="sect:accesstbl"/> for that.
Once we have such a character table,
we can compute all those data about the underlying group <M>G</M>, say,
that are determined by the character table.
Chapter <Ref Chap="Attributes and Properties for Groups and Character Tables"
BookName="ref"/> lists such attributes and properties.
For example, it can be computed from the character table of <M>G</M>
whether <M>G</M> is solvable or not.
<P/>
Questions that cannot be answered using only the character table of <M>G</M>
can perhaps be treated using additional information.
For example, the structure of subgroups of <M>G</M> is in general not
determined by the character table of <M>G</M>,
but the character table may yield partial information.
Two examples can be found in the sections
<Ref Subsect="subsect:sylowstructure3on"/> and
<Ref Subsect="subsect:permcharfi23"/>.
<P/>
In the character table context,
the role of homomorphisms between two groups is taken by
<E>class fusions</E>.
Monomorphisms correspond to subgroup fusions,
epimorphisms correspond to factor fusions.
Given two character tables of a group <M>G</M> and a subgroup <M>H</M>
of <M>G</M>,
one can in general compute only <E>candidates</E> for the class fusion of
<M>H</M> into <M>G</M>,
for example using <Ref Func="PossibleClassFusions" BookName="ref"/>.
Note that <M>G</M> may contain several nonconjugate subgroups isomorphic
with <M>H</M>, which may have different class fusions.
<!-- <P/>
If no subgroup fusion from the character table of the group <M>H</M> into the character table of the group <M>G</M> is possible then one has proved that <M>G</M> contains no subgroup isomorphic with <M>H</M>. If one knows that <M>G</M> contains subgroups isomorphic with <M>H</M> and if the class fusion between the character tables of these groups is unique (up to character table automorphisms) then this class fusion can be used to induce or restrict characters.
-->
<P/>
One can often reduce a question about a group <M>G</M> to a question about
its maximal subgroups.
In the character table context,
it is often sufficient to know the character table of <M>G</M>,
the character tables of its maximal subgroups,
and their class fusions into <M>G</M>.
We are in this situation if the attribute <Ref Attr="Maxes"/> is set in
the character table of <M>G</M>.
<P/>
<E>Summary:</E>
The character theoretic approach that is supported by the
&GAP; Character Table Library, that is, an approach without explicitly
using the underlying groups,
has the advantages that it can be used to answer many questions,
and that these computations are usually cheap,
compared to computations with groups.
Disadvantages are that this approach is not always successful,
and that answers are often <Q>nonconstructive</Q> in the sense that one
can show the existence of something without getting one's hands on it.
</Section>
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<Section Label="sect:accesstbl">
<Heading>Accessing a Character Table from the Library</Heading>
As stated in Section <Ref Sect="sect:concepts"/>,
we must define how character tables from the &GAP; Character Table Library
can be accessed.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:accesstblbyname">
<Heading>Accessing a Character Table via a name</Heading>
The most common way to access a character table from the
&GAP; Character Table Library is to call
<Ref Func="CharacterTable" Label="for a string"/> with argument
a string that is an <E>admissible name</E> for the character table.
Typical admissible names are similar to the group names used in the
&ATLAS; of Finite Groups <Cite Key="CCN85"/>.
One of these names is the
<Ref Attr="Identifier" Label="for character tables" BookName="ref"/> value
of the character table,
this name is used by &GAP; when it prints library character tables.
<P/>
For example, an admissible name for the character table of an
almost simple group is the &ATLAS; name,
such as <C>A5</C>, <C>M11</C>, or <C>L2(11).2</C>.
Other names may be admissible, for example <C>S6</C> is admissible for
the symmetric group on six points,
which is called <M>A_6.2_1</M> in the &ATLAS;.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:accesstblbyproperties">
<Heading>Accessing a Character Table via properties</Heading>
If one does not know an admissible name of the character table of a group
one is interested in, or if one does not know whether ths character table
is available at all,
one can use <Ref Func="AllCharacterTableNames"/> to compute a list of
identifiers of all available character tables with given properties.
Analogously, <Ref Func="OneCharacterTableName"/> can be used to compute
one such identifier.
For certain filters, such as <Ref Attr="Size" BookName="ref"/> and
<Ref Attr="NrConjugacyClasses" BookName="ref"/>,
the computations are fast because the values for all library tables
are precomputed.
See <Ref Func="AllCharacterTableNames"/> for an overview of these filters.
<P/>
The function <Ref Func="BrowseCTblLibInfo"/> provides an interactive overview
of available character tables,
which allows one for example to search also for substrings in identifiers of
character tables.
This function is available only if the <Package>Browse</Package> package
has been loaded.
</Subsection>
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<Subsection Label="subsect:accesstblbytom">
<Heading>Accessing a Character Table via a Table of Marks</Heading>
Let <M>G</M> be a group whose table of marks is available via the
<Package>TomLib</Package> package (see <Cite Key="TomLib"/> for how to access
tables of marks from this library)
then the &GAP; Character Table Library contains the character table of
<M>G</M>,
and one can access this table by using the table of marks as an argument of
<Ref Meth="CharacterTable" Label="for a table of marks"/>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:accesstblbytbl">
<Heading>Accessing a Character Table relative to another Character Table
</Heading>
If one has already a character table from the &GAP; Character Table Library
that belongs to the group <M>G</M>, say,
then names of related tables can be found as follows.
<P/>
The value of the attribute <Ref Attr="Maxes"/>, if known, is the list of
identifiers of the character tables of all classes of maximal subgroups
of <M>G</M>.
If the <Ref Attr="Maxes"/> value of the character table with identifier
<M>id</M>, say, is known then the character table of the groups in the
<M>i</M>-th class of maximal subgroups can be accessed via the
<Q>relative name</Q> <M>id</M><C>M</C><M>i</M>.
The value of the attribute <Ref Attr="NamesOfFusionSources" BookName="ref"/>
is the list of identifiers of those character tables which store
class fusions to <M>G</M>.
So these character tables belong to subgroups of <M>G</M>
and groups that have <M>G</M> as a factor group.
The value of the attribute <Ref Attr="ComputedClassFusions" BookName="ref"/>
is the list of records whose <C>name</C> components are the identifiers
of those character tables to which class fusions are stored.
So these character tables belong to overgroups and factor groups of <M>G</M>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:severaltables">
<Heading>Different character tables for the same group</Heading>
The &GAP; Character Table Library may contain several different
character tables of a given group,
in the sense that the rows and columns are sorted differently.
<P/>
For example, the &ATLAS; table of the alternating group <M>A_5</M> is
available, and since <M>A_5</M> is isomorphic with the groups
PSL<M>(2, 4)</M> and PSL<M>(2, 5)</M>, two more character tables of
<M>A_5</M> can be constructed in a natural way.
The three tables are of course permutation isomorphic.
The first two are sorted in the same way, but the rows and columns of the
third one are sorted differently.
Another situation where several character tables for the same group are
available is that a group contains several classes of isomorphic
maximal subgroups such that the class fusions are different.
<P/>
For example, the Mathieu group <M>M_{12}</M> contains two classes of
maximal subgroups of index <M>12</M>,
which are isomorphic with <M>M_{11}</M>.
The class fusions into <M>M_{12}</M> are stored on the library tables of the
maximal subgroups.
The groups in the first class of <M>M_{11}</M> type subgroups contain
elements in the classes <C>4B</C>, <C>6B</C>, and <C>8B</C> of <M>M_{12}</M>,
and the groups in the second class contain
elements in the classes <C>4A</C>, <C>6A</C>, and <C>8A</C>.
Note that according to the &ATLAS;
(see <Cite Key="CCN85" Where="p. 33"/>),
the permutation characters of the action of <M>M_{12}</M> on the cosets
of <M>M_{11}</M> type subgroups from the two classes of maximal subgroups
are <C>1a + 11a</C> and <C>1a + 11b</C>, respectively.
A = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
= (-1+Sqrt(-11))/2 = b11
]]></Example>
Permutation equivalent library tables are related to each other.
In the above example, the table <C>s2</C> is a <E>duplicate</E> of <C>s1</C>,
and there are functions for making the relations explicit.
See Section <Ref Sect="sec:duplicates"/> for details about
duplicate character tables.
</Subsection>
</Section>
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<Section Label="sect:tutsectctbllib">
<Heading>Examples of Using the &GAP; Character Table Library</Heading>
The sections <Ref Subsect="subsect:ambivalent"/>,
<Ref Subsect="subsect:ppure"/>, and
<Ref Subsect="subsect:onepblock"/> show how the function
<Ref Func="AllCharacterTableNames"/> can be used to search for
character tables with certain properties.
The &GAP; Character Table Library serves as a tool for finding and
checking conjectures in these examples.
<P/>
In Section <Ref Subsect="subsect:permcharfi23"/>,
a question about a subgroup of the sporadic simple Fischer group
<M>G = Fi_{23}</M> is answered using only character tables from the
&GAP; Character Table Library.
<P/>
More examples can be found in
<Cite Key="GMN"/>, <Cite Key="AmbigFus"/>, <Cite Key="ctblpope"/>,
<Cite Key="ProbGenArxiv"/>, <Cite Key="Auto"/>.
A group <M>G</M> is called <E>ambivalent</E> if each element in <M>G</M>
is <M>G</M>-conjugate to its inverse.
Equivalently, <M>G</M> is ambivalent if all its characters are real-valued.
We are interested in nonabelian simple ambivalent groups.
Since ambivalence is of course invariant under permutation equivalence,
we may omit duplicate character tables.
A group <M>G</M> is called <E><M>p</M>-pure</E> for a prime integer <M>p</M>
that divides <M>|G|</M>
if the centralizer orders of nonidentity <M>p</M>-elements in <M>G</M>
are <M>p</M>-powers.
Equivalently, <M>G</M> is <M>p</M>-pure if <M>p</M> divides <M>|G|</M>
and each element in <M>G</M> of order divisible by <M>p</M>
is a <M>p</M>-element.
(This property was studied by L. Héthelyi in 2002.)
<P/>
We are interested in small nonabelian simple <M>p</M>-pure groups.
Looking at these examples, we may observe that the alternating group
<M>A_n</M> of degree <M>n</M> is
<M>2</M>-pure iff <M>n \in \{ 4, 5, 6 \}</M>,
<M>3</M>-pure iff <M>n \in \{ 3, 4, 5, 6 \}</M>, and
<M>p</M>-pure, for <M>p \geq 5</M>, iff <M>n \in \{ p, p+1, p+2 \}</M>.
<P/>
Also, the Suzuki groups <M>Sz(q)</M> are <M>2</M>-pure
since the centralizers of nonidentity <M>2</M>-elements are contained in
Sylow <M>2</M>-subgroups.
<P/>
From the inspection of the generic character table(s) of <M>PSL(2, q)</M>,
we see that <M>PSL(2, p^d)</M> is <M>p</M>-pure
Additionally, exactly the following cases of <M>l</M>-purity occur,
for a prime <M>l</M>.
<List>
<Item>
<M>q</M> is even and <M>q-1</M> or <M>q+1</M> is a power of <M>l</M>.
</Item>
<Item>
For <M>q \equiv 1 \pmod{4}</M>, <M>(q+1)/2</M> is a power of <M>l</M>
or <M>q-1</M> is a power of <M>l = 2</M>.
</Item>
<Item>
For <M>q \equiv 3 \pmod{4}</M>, <M>(q-1)/2</M> is a power of <M>l</M>
or <M>q+1</M> is a power of <M>l = 2</M>.
</Item>
</List>
<!-- somehow, the special case L3(2) came up in this context; there is a paper by Bob Guralnick on subgroups of prime power index in simple groups
-->
</Subsection>
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<Subsection Label="subsect:onepblock">
<Heading>Example: Simple Groups with only one <M>p</M>-Block</Heading>
Are there nonabelian simple groups with only one <M>p</M>-block,
for some prime <M>p</M>?
<Example><![CDATA[
gap> fun:= function( tbl )
> local result, p, bl;
>
> result:= false;
> for p in PrimeDivisors( Size( tbl ) ) do
> bl:= PrimeBlocks( tbl, p );
> if Length( bl.defect ) = 1 then
> result:= true;
> Print( "only one block: ", Identifier( tbl ), ", p = ", p, "\n" );
> fi;
> od;
>
> return result;
> end;;
gap> AllCharacterTableNames( IsSimple, true, IsAbelian, false,
> IsDuplicateTable, false, fun, true );
only one block: M22, p = 2
only one block: M24, p = 2
[ "M22", "M24" ]
]]></Example>
We see that the sporadic simple groups <M>M_{22}</M> and <M>M_{24}</M>
have only one <M>2</M>-block.
<!-- for alternating groups, see James/Kerber; Lie groups in def. characteristics do always have a Steinberg character, so there must be at least two blocks
-->
We want to determine the structure of the Sylow <M>3</M>-subgroups of the
triple cover <M>G = 3.O'N of the sporadic simple O'Nan group <M>O'N.
The Sylow <M>3</M>-subgroup of <M>O'N is an elementary abelian group
of order <M>3^4</M>,
since the Sylow <M>3</M> normalizer in <M>O'N has the structure
<M>3^4:2^{1+4}D_{10}</M> (see <Cite Key="CCN85" Where="p. 132"/>).
Let <M>P</M> be a Sylow <M>3</M>-subgroup of <M>G</M>.
Then <M>P</M> is not abelian,
since the centralizer order of any preimage of an element of order three
in the simple factor group of <M>G</M> is not divisible by <M>3^5</M>.
Moreover, the exponent of <M>P</M> is three.
So both the centre and the Frattini subgroup of <M>P</M> are equal to the
centre of <M>G</M>,
hence <M>P</M> is an extraspecial group <M>3^{1+4}_+</M>.
It is often interesting to compute the primitive permutation characters
of a group <M>G</M>,
that is, the characters of the permutation actions of <M>G</M> on the cosets
of its maximal subgroups.
These characters can be computed for example when the character tables of
<M>G</M>, the character tables of its maximal subgroups,
and the class fusions from these character tables into the table of <M>G</M>
are known.
We see that the two approaches yield the same permutation characters,
but the two lists are sorted in a different way.
The latter is due to the fact that the rows of the table of marks are
ordered in a way that is not compatible with the ordering of maximal
subgroups for the character table.
Moreover, there is no way to choose the fusion from the character table
to the table of marks in such a way that the two lists of
permutation characters would become equal.
The component <C>perm</C> in the <Ref Attr="FusionToTom"/> record of
the character table describes the incompatibility.
<Example><![CDATA[
gap> FusionToTom( tbl );
rec( map := [ 1, 2, 5, 4, 8, 3, 7, 11, 11, 6, 13, 6, 13 ],
name := "2.A6", perm := (4,5),
text := "fusion map is unique up to table autom." )
]]></Example>
</Subsection>
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<Subsection Label="subsect:permcharfi23">
<Heading>Example: A Permutation Character of <M>Fi_{23}</M></Heading>
Let <M>x</M> be a <C>3B</C> element in the sporadic simple Fischer group
<M>G = Fi_{23}</M>.
The normalizer <M>M</M> of <M>x</M> in <M>G</M> is a maximal subgroup
of the type <M>3^{{1+8}}_+.2^{{1+6}}_-.3^{{1+2}}_+.2S_4</M>.
We are interested in the distribution of the elements of the
normal subgroup <M>N</M> of the type <M>3^{{1+8}}_+</M> in <M>M</M>
to the conjugacy classes of <M>G</M>.
<P/>
This information can be computed from the permutation character
<M>\pi = 1_N^G</M>, so we try to compute this permutation character.
We have <M>\pi = (1_N^M)^G</M>,
and <M>1_N^M</M> can be computed as the inflation of
the regular character of the factor group <M>M/N</M> to <M>M</M>.
Note that the character tables of <M>G</M> and <M>M</M> are available,
as well as the class fusion of <M>M</M> in <M>G</M>,
and that <M>N</M> is the largest normal <M>3</M>-subgroup of <M>M</M>.
Thus <M>N</M> contains <M>864</M> elements in the class <C>3A</C>,
<M>1\,538</M> elements in the class <C>3B</C>, and so on.
</Subsection>
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<Subsection Label="subsect:commutatorlength">
<Heading>Example: Non-commutators in the commutator group</Heading>
In general, not every element in the commutator group of a group is itself
a commutator.
Are there examples in the Character Table Library,
and if yes, what is a smallest one?
<Example><![CDATA[
gap> nam:= OneCharacterTableName( CommutatorLength, x -> x > 1
> : OrderedBy:= Size ); "3.(A4x3):2"
gap> Size( CharacterTable( nam ) );
216
]]></Example>
The smallest groups with this property have order <M>96</M>.
<Example><![CDATA[
gap> OneSmallGroup( Size, [ 2 .. 100 ],
> G -> CommutatorLength( G ) > 1, true );
<pc group of size 96 with 6 generators>
]]></Example>
(Note the different syntax:
<Ref Func="OneSmallGroup" BookName="smallgrp"/> does not admit a function
such as <C>x -> x > 1</C> for describing the admissible values.)
<P/>
Nonabelian simple groups cannot be expected to have non-commutators,
by the main theorem in <Cite Key="LOST2010"/>.
<Example><![CDATA[
gap> nam:= OneCharacterTableName( IsPerfect, true,
> IsDuplicateTable, false,
> CommutatorLength, x -> x > 1
> : OrderedBy:= Size ); "P1/G1/L1/V1/ext2"
gap> Size( CharacterTable( nam ) );
960
]]></Example>
This is in fact the smallest example of a perfect group that contains
non-commutators.
<Example><![CDATA[
gap> for n in [ 2 .. 960 ] do
> for i in [ 1 .. NrPerfectGroups( n ) ] do
> g:= PerfectGroup( n, i);
> if CommutatorLength( g ) <> 1 then
> Print( [ n, i ], "\n" );
> fi;
> od;
> od;
[ 960, 2 ]
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:J4mod11deg887778">
<Heading>Example: An irreducible <M>11</M>-modular character of <M>J_4</M>
(December 2018)</Heading>
Let <M>G</M> be the sporadic simple Janko group <M>J_4</M>.
For the ordinary irreducible characters of degree <M>1333</M> of <M>G</M>,
the reductions modulo <M>11</M> are known to be irreducible Brauer characters.
<P/>
David Craven asked Richard Parker how to show that the antisymmetric squares
of these Brauer characters are irreducible.
Richard proposed the following.
<P/>
Restrict the given ordinary character <M>\chi</M>, say,
to a subgroup <M>S</M> of <M>J_4</M>
whose <M>11</M>-modular character table is known,
decompose the restriction <M>\chi_S</M> into irreducible Brauer characters,
and compute those constituents that are constant on all subsets of
conjugacy classes that fuse in <M>J_4</M>.
If the Brauer character <M>\chi_S</M> cannot be written as a sum
of two such constituents then <M>\chi</M>,
as a Brauer character of <M>J_4</M>, is irreducible.
<P/>
Here is a &GAP; session that shows how to apply this idea.
<P/>
The group <M>J_4</M> has exactly two ordinary irreducible characters of
degree <M>1333</M>.
They are complex conjugate, and so are their antisymmetric squares.
Thus we may consider just one of the two.
Let <M>S</M> be a maximal subgroup of the structure <M>2^{11}:M_{24}</M>
in <M>J_4</M>.
Fortunately,
the <M>11</M>-modular character table of <M>S</M> is available
(it had been constructed by Christoph Jansen),
and we can restrict the interesting character to this table.
<Example><![CDATA[
gap> s:= CharacterTable( Maxes( t )[1] );;
gap> Size( s ) = 2^11 * Size( CharacterTable( "M24" ) );
true
gap> rest:= RestrictedClassFunction( chi, s );;
gap> smod11:= s mod 11;;
gap> rest:= RestrictedClassFunction( rest, smod11 );;
]]></Example>
The restriction is a sum of nine pairwise different irreducible
Brauer characters of <M>S</M>.
We see that no proper subset of the constituents yields a Brauer character
that can be restricted from <M>J_4</M>.
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:nonprojtensor">
<Heading>Example: Tensor Products that are Generalized Projectives
(October 2019)</Heading>
Let <M>G</M> be a finite group and <M>p</M> be a prime integer.
If the tensor product <M>\Phi</M>, say,
of two ordinary irreducible characters of <M>G</M>
vanishes on all <M>p</M>-singular elements of <M>G</M> then
<M>\Phi</M> is a <M>&ZZ;</M>-linear combination of the
<E>projective indecomposable characters</E>
<M>\Phi_{\varphi} = \sum_{{\chi \in &Irr;(G)}} d_{{\chi \varphi}} \chi</M>
of <M>G</M>,
where <M>\varphi</M> runs over the irreducible <M>p</M>-modular
Brauer characters of <M>G</M> and <M>d_{{\chi \varphi}}</M> is the
decomposition number of <M>\chi</M> and <M>\varphi</M>.
(See for example <Cite Key="Nav98" Where="p. 25"/> or
<Cite Key="LP10" Where="Def. 4.3.1"/>.)
Such class functions are called generalized projective characters.
<P/>
In fact, very often <M>\Phi</M> is a projective character, that is,
the coefficients of the decomposition into
projective indecomposable characters are nonnegative.
<P/>
We are interested in examples where this is <E>not</E> the case.
For that, we write a small &GAP; function that computes,
for a given <M>p</M>-modular character table,
those tensor products of ordinary irreducible characters
that are generalized projective characters but are not projective.
<P/>
Many years ago, Richard Parker had been interested in the question
whether such tensor products can exist for a given group.
Note that forming tensor products that vanish on <M>p</M>-singular elements
is a recipe for creating projective characters,
provided one knows in advance that the answer is negative for the
given group.
<P/>
<Example><![CDATA[
gap> GenProjNotProj:= function( modtbl )
> local p, tbl, X, PIMs, n, psingular, list, labels, i, j, psi,
> pos, dec, poss;
>
> p:= UnderlyingCharacteristic( modtbl );
> tbl:= OrdinaryCharacterTable( modtbl );
> X:= Irr( tbl );
> PIMs:= TransposedMat( DecompositionMatrix( modtbl ) ) * X;
> n:= Length( X );
> psingular:= Difference( [ 1 .. n ], GetFusionMap( modtbl, tbl ) );
> list:= [];
> labels:= [];
> for i in [ 1 .. n ] do
> for j in [ 1 .. i ] do
> psi:= List( [ 1 .. n ], x -> X[i][x] * X[j][x] );
> if IsZero( psi{ psingular } ) then
> # This is a generalized projective character.
> pos:= Position( list, psi );
> if pos = fail then
> Add( list, psi );
> Add( labels, [ [ j, i ] ] );
> else
> Add( labels[ pos ], [ j, i ] );
> fi;
> fi;
> od;
> od;
>
> if Length( list ) > 0 then
> # Decompose the generalized projective tensor products
> # into the projective indecomposables.
> dec:= Decomposition( PIMs, list, "nonnegative" );
> poss:= Positions( dec, fail );
> return Set( Concatenation( labels{ poss } ) );
> else
> return [];
> fi;
> end;;
]]></Example>
<P/>
One group for which the function returns a nonempty result is the sporadic
simple Janko group <M>J_2</M> in characteristic <M>2</M>.
Checking all available tables from the library takes several hours of
CPU time and also requires a lot of space; <!-- "6x2.F4(2)" needs more than 7GB -->
finally, it yields the following result.
<P/>
<Example><![CDATA[
gap> examples:= [];;
gap> for name in AllCharacterTableNames( IsDuplicateTable, false ) do
> tbl:= CharacterTable( name );
> for p in PrimeDivisors( Size( tbl ) ) do
> modtbl:= tbl mod p;
> if modtbl <> fail then
> res:= GenProjNotProj( modtbl );
> if not IsEmpty( res ) then
> AddSet( examples, [ name, p, Length( res ) ] );
> fi;
> fi;
> od;
> od;
gap> examples;
[ [ "(A5xJ2):2", 2, 4 ], [ "(D10xJ2).2", 2, 9 ], [ "2.Suz", 3, 1 ],
[ "2.Suz.2", 3, 4 ], [ "2xCo2", 5, 4 ], [ "3.Suz", 2, 6 ],
[ "3.Suz.2", 2, 4 ], [ "Co2", 5, 1 ], [ "Co3", 2, 4 ],
[ "Isoclinic(2.Suz.2)", 3, 4 ], [ "J2", 2, 1 ], [ "Suz", 2, 2 ],
[ "Suz", 3, 1 ], [ "Suz.2", 3, 4 ] ]
]]></Example>
This list looks rather <Q>sporadic</Q>.
The number of examples is small, and all groups in question except two
(the subdirect products of <M>S_5</M> and <M>J_2.2</M>,
and of <M>5:4</M> and <M>J_2.2</M>, respectively)
are extensions of sporadic simple groups.
<P/>
Note that the following cases could be omitted
because the characters in question belong to proper factor groups:
<M>2.Suz</M> mod <M>3</M>, <M>2.Suz.2</M> mod <M>3</M>,
and its isoclinic variant.
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:elabsubgroup">
<Heading>Example: Certain elementary abelian subgroups in quasisimple groups
(November 2020)</Heading>
In October 2020, Bob Guralnick asked:
Does each quasisimple group <M>G</M> contain an elementary abelian
subgroup that contains elements from all conjugacy classes
of involutions in <M>G</M>?
(Such a subgroup is called a <E>broad</E> subgroup of <M>G</M>.
See <Cite Key="GR20"/> for the paper.)
<P/>
In the case of simple groups,
theoretical arguments suffice to show that the answer is positive for
simple groups of alternating and Lie type,
thus it remains to inspect the sporadic simple groups.
<P/>
In the case of nonsimple quasisimple groups,
again groups having a sporadic simple factor group have to be checked,
and also the central extensions of groups of Lie type by exceptional
multipliers have to be checked computationally.
<P/>
In the following situations,
the answer is positive for a given group <M>G</M>.
<P/>
<Enum>
<Item>
<M>G</M> has at most two classes of involutions.
(Take an involution <M>x</M> in the centre of a Sylow <M>2</M>-subgroup
<M>P</M> of <M>G</M>;
if there is a conjugacy class of involutions in <M>G</M>
different from <M>x^G</M> then <M>P</M> contains an element in the other
involution class.)
</Item>
<Item>
<M>G</M> has exactly three classes of involutions
such that there are representatives <M>x</M>, <M>y</M>, <M>z</M>
with the property <M>x y = z</M>.
(The subgroup <M>\langle x, y \rangle</M> is a Klein four group;
note that <M>x</M> and <M>y</M> commute because
<M>x^{{-1}} y^{{-1}} x y = (x y)^2 = z^2 = 1</M> holds.)
</Item>
<Item>
<M>G</M> has a central elementary abelian <M>2</M>-subgroup <M>N</M>,
and there is an elementary abelian <M>2</M>-subgroup <M>P / N</M> in
<M>G / N</M> containing elements from all those involution classes of
<M>G / N</M> that lift to involutions of <M>G</M>, but no elements from
other involution classes of <M>G / N</M>.
(Just take the preimage <M>P</M>, which is elementary abelian.)
<P/>
This condition is satisfied for example if the answer is positive for
<M>G / N</M> and <E>all</E> involutions of <M>G / N</M> lift to
involutions in <M>G</M>,
or if exactly one class of involutions of <M>G / N</M> lifts to
involutions in <M>G</M>.
</Item>
</Enum>
<P/>
The following function evaluates the first two of the above criteria
and easy cases of the third one, for the given character table
of the group <M>G</M>.
<P/>
<Example><![CDATA[
gap> ApplyCriteria:= "dummy";; # Avoid a syntax error ...
gap> ApplyCriteria:= function( tbl )
> local id, ord, invpos, cen, facttbl, factfus, invmap, factord,
> factinvpos, imgs;
> id:= ReplacedString( Identifier( tbl ), " ", "" );
> ord:= OrdersClassRepresentatives( tbl );
> invpos:= PositionsProperty( ord, x -> x <= 2 );
> if Length( invpos ) <= 3 then
> # There are at most 2 involution classes.
> Print( "#I ", id, ": ",
> "done (", Length( invpos ) - 1, " inv. class(es))\n" );
> return true;
> elif Length( invpos ) = 4 and
> ClassMultiplicationCoefficient( tbl, invpos[2], invpos[3],
> invpos[4] ) <> 0 then
> Print( "#I ", id, ": ",
> "done (3 inv. classes, nonzero str. const.)\n" );
> return true;
> fi;
> cen:= Intersection( invpos, ClassPositionsOfCentre( tbl ) );
> if Length( cen ) > 1 then
> # Consider the factor modulo the largest central el. ab. 2-group.
> facttbl:= tbl / cen;
> factfus:= GetFusionMap( tbl, facttbl );
> invmap:= InverseMap( factfus );
> factord:= OrdersClassRepresentatives( facttbl );
> factinvpos:= PositionsProperty( factord, x -> x <= 2 );
> if ForAll( factinvpos,
> i -> invmap[i] in invpos or
> ( IsList( invmap[i] ) and
> IsSubset( invpos, invmap[i] ) ) ) then
> # All involutions of the factor group lift to involutions.
> if ApplyCriteria( facttbl ) = true then
> Print( "#I ", id, ": ",
> "done (all inv. in ",
> ReplacedString( Identifier( facttbl ), " ", "" ),
> " lift to inv.)\n" );
> return true;
> fi;
> fi;
> imgs:= Set( factfus{ invpos } );
> if Length( imgs ) = 2 and
> ForAll( imgs,
> i -> invmap[i] in invpos or
> ( IsList( invmap[i] ) and
> IsSubset( invpos, invmap[i] ) ) ) then
> # There is a C2 subgroup of the factor
> # such that its involution lifts to involutions,
> # and the lifts of the C2 cover all involution classes of 'tbl'.
> Print( "#I ", id, ": ",
> "done (all inv. in ", id,
> " are lifts of a C2\n",
> "#I in the factor modulo ",
> ReplacedString( String( cen ), " ", "" ), ")\n" );
> return true;
> fi;
> fi;
> Print( "#I ", id, ": ",
> "OPEN (", Length( invpos ) - 1, " inv. class(es))\n" );
> return false;
> end;;
]]></Example>
The group <M>G = B</M> contains maximal subgroups of the type
<M>5:4 \times HS.2</M> (the normalizers of <C>5A</C> elements,
see <Cite Key="CCN85" Where="p. 217"/>).
The direct factor <M>H = HS.2</M> of such a subgroup
has four classes of involutions,
which fuse to the four involution classes of <M>G</M>.
The table of marks of <M>H</M> is known.
We find five classes of elementary abelian subgroups of order eight
in <M>H</M> that contain elements from all four involution classes of
<M>H</M>.
<P/>
<Example><![CDATA[
gap> tom:= TableOfMarks( h );
TableOfMarks( "HS.2" )
gap> ord:= OrdersTom( tom );;
gap> invpos:= Positions( ord, 2 );
[ 2, 3, 534, 535 ]
gap> 8pos:= Positions( ord, 8 );;
gap> filt:= Filtered( 8pos,
> x -> ForAll( invpos,
> y -> Length( IntersectionsTom( tom, x, y ) ) >= y
> and IntersectionsTom( tom, x, y )[y] <> 0 ) );
[ 587, 589, 590, 593, 595 ]
gap> reps:= List( filt, i -> RepresentativeTom( tom, i ) );;
gap> ForAll( reps, IsElementaryAbelian );
true
]]></Example>
<P/>
The group <M>G = Co_1</M> has a maximal subgroup <M>H</M> of type
<M>A_9 \times S_3</M>
(see <Cite Key="CCN85" Where="p. 183"/>)
that contains elements from all three involution classes of <M>G</M>.
Moreover, the factor <M>S_3</M> contains <C>2A</C> elements,
and the factor <M>A_9</M> contains <C>2B</C> and <C>2C</C> elements.
This yields the desired elementary abelian subgroup of order eight.
<!-- alternative arguments for some sporadic groups (not needed) <P/>
The group <M>G = Co_2</M> has a maximal subgroup <M>H</M> of type <M>HS.2</M> that contains elements from all classes of involutions in <M>G</M>. As we have seen in the case of the group <M>B</M>, there are elementary abelian subgroups of order eight in <M>H</M> that contain elements from all classes of involutions in <M>H</M> and hence also elements from all three involution classes in <M>G</M>.
For <M>G = Fi_{22}</M>, consider the maximal subgroup <M>2^{10}.M_{22}</M> of <M>G</M>: Its <M>2</M>-core is elementary abelian and covers all three involution classes of <M>G</M>.
Thus we know that the answer is positive for each sporadic simple group.
Next we look at the relevant covering groups of sporadic simple groups.
For a quasisimple group with a sporadic simple factor,
the Schur multiplier has at most the prime factors <M>2</M> and <M>3</M>;
only the extension by the <M>2</M>-part of the multipier must be checked.
The group <M>B</M> has four classes of involutions,
let us call them <C>2A</C>, <C>2B</C>, <C>2C</C>, and <C>2D</C>.
All except <C>2C</C> lift to involutions in <M>2.B</M>.
Thus it suffices to show that there is a subgroup of type <M>2^2</M>
in <M>B</M> that contains elements from <C>2A</C>, <C>2B</C>, and <C>2D</C>
(but no element from <C>2C</C>).
This follows from the fact that the
<M>(</M><C>2A</C>, <C>2B</C>, <C>2D</C><M>)</M> structure constant
of <M>B</M> is nonzero.
The group <M>Co_1</M> has three classes of involutions,
let us call them <C>2A</C>, <C>2B</C>, and <C>2C</C>.
All except <C>2B</C> lift to involutions in <M>2.Co_1</M>.
Thus it suffices to show that there is a subgroup of type <M>2^2</M>
in <M>Co_1</M> that contains elements from <C>2A</C> and <C>2C</C>
but no element from <C>2B</C>.
This follows from the fact that the
<M>(</M><C>2A</C>, <C>2A</C>, <C>2C</C><M>)</M> structure constant
of <M>Co_1</M> is nonzero.
Finally,
we deal with the relevant central extensions of finite simple groups
of Lie type with exceptional multipliers.
These groups are listed in <Cite Key="CCN85" Where="p. xvi, Table 5"/>.
The following cases belong to exceptional multipliers with nontrivial
<M>2</M>-part.
This leads to the following list of cases to be checked.
(We would not need to deal with the groups <M>A_5</M> and <M>L_3(2)</M>,
because of isomorphisms with groups of Lie type for which
the multiplier in question is not exceptional, but here we ignore this fact.)
<P/>
<Example><![CDATA[
gap> list:= [
> [ "A5", "2.A5" ],
> [ "L3(2)", "2.L3(2)" ],
> [ "L3(4)", "2.L3(4)", "2^2.L3(4)", "4_1.L3(4)", "4_2.L3(4)",
> "(2x4).L3(4)", "4^2.L3(4)" ],
> [ "A8", "2.A8" ],
> [ "U4(2)", "2.U4(2)"],
> [ "U6(2)", "2.U6(2)", "2^2.U6(2)" ],
> [ "A6", "2.A6" ],
> [ "Sz(8)", "2.Sz(8)", "2^2.Sz(8)" ],
> [ "S6(2)", "2.S6(2)" ],
> [ "O8+(2)", "2.O8+(2)", "2^2.O8+(2)" ],
> [ "G2(4)", "2.G2(4)" ],
> [ "F4(2)", "2.F4(2)" ],
> [ "2E6(2)", "2.2E6(2)", "2^2.2E6(2)" ] ];;
gap> Filtered( Concatenation( list ),
> x -> not ApplyCriteria( CharacterTable( x ) ) );
#I A5: done (1 inv. class(es))
#I 2.A5: done (1 inv. class(es))
#I L3(2): done (1 inv. class(es))
#I 2.L3(2): done (1 inv. class(es))
#I L3(4): done (1 inv. class(es))
#I 2.L3(4): done (3 inv. classes, nonzero str. const.)
#I 2^2.L3(4)/[1,2,3,4]: done (1 inv. class(es))
#I 2^2.L3(4): done (all inv. in 2^2.L3(4)/[1,2,3,4] lift to inv.)
#I 4_1.L3(4): done (2 inv. class(es))
#I 4_2.L3(4): done (2 inv. class(es))
#I (2x4).L3(4): done (all inv. in (2x4).L3(4) are lifts of a C2
#I in the factor modulo [1,2,3,4])
#I 4^2.L3(4): done (all inv. in 4^2.L3(4) are lifts of a C2
#I in the factor modulo [1,2,3,4])
#I A8: done (2 inv. class(es))
#I 2.A8: done (2 inv. class(es))
#I U4(2): done (2 inv. class(es))
#I 2.U4(2): done (2 inv. class(es))
#I U6(2): done (3 inv. classes, nonzero str. const.)
#I 2.U6(2)/[1,2]: done (3 inv. classes, nonzero str. const.)
#I 2.U6(2): done (all inv. in 2.U6(2)/[1,2] lift to inv.)
#I 2^2.U6(2)/[1,2,3,4]: done (3 inv. classes, nonzero str. const.)
#I 2^2.U6(2): done (all inv. in 2^2.U6(2)/[1,2,3,4] lift to inv.)
#I A6: done (1 inv. class(es))
#I 2.A6: done (1 inv. class(es))
#I Sz(8): done (1 inv. class(es))
#I 2.Sz(8): done (2 inv. class(es))
#I 2^2.Sz(8)/[1,2,3,4]: done (1 inv. class(es))
#I 2^2.Sz(8): done (all inv. in 2^2.Sz(8)/[1,2,3,4] lift to inv.)
#I S6(2): OPEN (4 inv. class(es))
#I 2.S6(2): OPEN (3 inv. class(es))
#I O8+(2): OPEN (5 inv. class(es))
#I 2.O8+(2): OPEN (5 inv. class(es))
#I 2^2.O8+(2): OPEN (5 inv. class(es))
#I G2(4): done (2 inv. class(es))
#I 2.G2(4): done (3 inv. classes, nonzero str. const.)
#I F4(2): OPEN (4 inv. class(es))
#I 2.F4(2)/[1,2]: OPEN (4 inv. class(es))
#I 2.F4(2): OPEN (9 inv. class(es))
#I 2E6(2): done (3 inv. classes, nonzero str. const.)
#I 2.2E6(2)/[1,2]: done (3 inv. classes, nonzero str. const.)
#I 2.2E6(2): done (all inv. in 2.2E6(2)/[1,2] lift to inv.)
#I 2^2.2E6(2)/[1,2,3,4]: done (3 inv. classes, nonzero str. const.)
#I 2^2.2E6(2): done (all inv. in 2^2.2E6(2)/[1,2,3,4] lift to inv.)
[ "S6(2)", "2.S6(2)", "O8+(2)", "2.O8+(2)", "2^2.O8+(2)", "F4(2)", "2.F4(2)" ]
]]></Example>
<P/>
We could assume that the answer is positive for the simple groups in
the list of open cases, by theoretical arguments,
but it is easy to show this computationally.
<P/>
For <M>G = S_6(2)</M>, consider a maximal subgroup <M>2^6.L_3(2)</M>
of <M>G</M> (see <Cite Key="CCN85" Where="p. 46"/>):
Its <M>2</M>-core is elementary abelian and covers all four
involution classes of <M>G</M>.
Concerning <M>G = 2.S_6(2)</M>,
note that from the four involution classes of <M>S_6(2)</M>,
exactly <C>2B</C> and <C>2D</C> lift to involutions in <M>2.S_6(2)</M>.
Thus it suffices to show that there is a subgroup of type <M>2^2</M>
in <M>S_6(2)</M> that contains elements from <C>2B</C> and <C>2D</C>
but no elements from <C>2A</C> or <C>2C</C>.
This follows from the fact that the
<M>(</M><C>2B</C>, <C>2D</C>, <C>2D</C><M>)</M> structure constant
of <M>S_6(2)</M> is nonzero.
For <M>G = O_8^+(2)</M>, we consider the known table of marks.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "O8+(2)" );;
gap> tom:= TableOfMarks( t );
TableOfMarks( "O8+(2)" )
gap> ord:= OrdersTom( tom );;
gap> invpos:= Positions( ord, 2 );
[ 2, 3, 4, 5, 6 ]
gap> 8pos:= Positions( ord, 8 );;
gap> filt:= Filtered( 8pos,
> x -> ForAll( invpos,
> y -> Length( IntersectionsTom( tom, x, y ) ) >= y
> and IntersectionsTom( tom, x, y )[y] <> 0 ) );
[ 151, 153 ]
gap> reps:= List( filt, i -> RepresentativeTom( tom, i ) );;
gap> ForAll( reps, IsElementaryAbelian );
true
]]></Example>
<P/>
Concerning <M>G = 2.O_8^+(2)</M>,
note that from the five involution classes of <M>O_8^+(2)</M>,
exactly <C>2A</C>, <C>2B</C>, and <C>2E</C> lift to involutions
in <M>2.O_8^+(2)</M>.
Concerning <M>G = 2^2.O_8^+(2)</M>,
note that from the five involution classes of <M>O_8^+(2)</M>,
exactly the first and the last lift to involutions
in <M>2^2.O_8^+(2)</M>.
For <M>G = F_4(2)</M>, consider a maximal subgroup <M>2^{10}.A_8</M>
of a maximal subgroup <M>S_8(2)</M> of <M>G</M>
(see <Cite Key="CCN85" Where="p. 123 and 170"/>):
Its <M>2</M>-core is elementary abelian and covers all four
involution classes of <M>G</M>.
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