  
  [1X75 [33X[0;0YMonomiality Questions[133X[101X
  
  [33X[0;0YThis  chapter  describes  functions  dealing  with the monomiality of finite
  (solvable) groups and their characters.[133X
  
  [33X[0;0YAll  these  functions  assume  [13Xcharacters[113X  to  be  class function objects as
  described in Chapter [14X72[114X, lists of character [13Xvalues[113X are not allowed.[133X
  
  [33X[0;0YThe  usual  [13Xproperty  tests[113X  of [5XGAP[105X that return either [9Xtrue[109X or [9Xfalse[109X are not
  sufficient  for  us.  When  we ask whether a group character [22Xχ[122X has a certain
  property,  such  as  quasiprimitivity, we usually want more information than
  just  yes or no. Often we are interested in the reason [13Xwhy[113X a group character
  [22Xχ[122X  was proved to have a certain property, e.g., whether monomiality of [22Xχ[122X was
  proved by the observation that the underlying group is nilpotent, or whether
  it  was necessary to construct a linear character of a subgroup from which [22Xχ[122X
  can  be induced. In the latter case we also may be interested in this linear
  character.  Therefore we need test functions that return a record containing
  such  useful  information.  For example, the record returned by the function
  [2XTestQuasiPrimitive[102X  ([14X75.3-3[114X)  contains the component [10XisQuasiPrimitive[110X (which
  is the known boolean property flag), and additionally the component [10Xcomment[110X,
  a string telling the reason for the value of the [10XisQuasiPrimitive[110X component,
  and  in  the  case  that  the  argument  [22Xχ[122X  was  [13Xnot[113X quasiprimitive also the
  component [10Xcharacter[110X, which is an irreducible constituent of a nonhomogeneous
  restriction  of  [22Xχ[122X  to a normal subgroup. Besides these test functions there
  are  also the known properties, e.g., the property [2XIsQuasiPrimitive[102X ([14X75.3-3[114X)
  which  will  call  the attribute [2XTestQuasiPrimitive[102X ([14X75.3-3[114X), and return the
  value of the [10XisQuasiPrimitive[110X component of the result.[133X
  
  [33X[0;0YA few words about how to use the monomiality functions seem to be necessary.
  Monomiality  questions  usually  involve  computations in many subgroups and
  factor  groups  of  a  given  group,  and  for  these groups often expensive
  calculations  such  as  that  of  the  character table are necessary. So one
  should  be  careful  not  to  construct  the same group over and over again,
  instead  the  same  group  object  should be reused, such that its character
  table  need  to  be  computed  only  once.  For example, suppose you want to
  restrict a character to a normal subgroup [22XN[122X that was constructed as a normal
  closure  of  some group elements, and suppose that you have already computed
  with   normal   subgroups   (by   calls   to  [2XNormalSubgroups[102X  ([14X39.19-9[114X)  or
  [2XMaximalNormalSubgroups[102X  ([14X39.19-10[114X))  and  their  character  tables. Then you
  should  look  in the lists of known normal subgroups whether [22XN[122X is contained,
  and  if  so you can use the known character table. A mechanism that supports
  this for normal subgroups is described in [14X71.23[114X.[133X
  
  [33X[0;0YAlso  the  following  hint  may  be useful in this context. If you know that
  sooner  or  later  you will compute the character table of a group [22XG[122X then it
  may be advisable to compute it as soon as possible. For example, if you need
  the  normal subgroups of [22XG[122X then they can be computed more efficiently if the
  character  table  of  [22XG[122X  is  known, and they can be stored compatibly to the
  contained  [22XG[122X-conjugacy  classes.  This  correspondence  of  classes list and
  normal subgroup can be used very often.[133X
  
  [33X[0;0YSeveral [13Xexamples[113X in this chapter use the symmetric group [22XS_4[122X and the special
  linear  group  [22XSL(2,3)[122X.  For running the examples, you must first define the
  groups, for example as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS4:= SymmetricGroup( 4 );;  SetName( S4, "S4" );[127X[104X
    [4X[25Xgap>[125X [27XSl23:= SL( 2, 3 );;[127X[104X
  [4X[32X[104X
  
  
  [1X75.1 [33X[0;0YInfoMonomial (Info Class)[133X[101X
  
  [1X75.1-1 InfoMonomial[101X
  
  [33X[1;0Y[29X[2XInfoMonomial[102X [32X info class[133X
  
  [33X[0;0YMost  of  the  functions  described  in  this  chapter print some (hopefully
  useful)  [13Xinformation[113X  if the info level of the info class [2XInfoMonomial[102X is at
  least [22X1[122X, see [14X7.4[114X for details.[133X
  
  
  [1X75.2 [33X[0;0YCharacter Degrees and Derived Length[133X[101X
  
  [1X75.2-1 Alpha[101X
  
  [33X[1;0Y[29X[2XAlpha[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a group [3XG[103X, [2XAlpha[102X returns a list whose [22Xi[122X-th entry is the maximal derived
  length  of  groups  [22X[3XG[103X  /  ker(χ)[122X  for  [22Xχ ∈ Irr([3XG[103X)[122X with [22Xχ(1)[122X at most the [22Xi[122X-th
  irreducible degree of [3XG[103X.[133X
  
  [1X75.2-2 Delta[101X
  
  [33X[1;0Y[29X[2XDelta[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  group  [3XG[103X, [2XDelta[102X returns the list [22X[ 1, alp[2] - alp[1], ..., alp[[3Xn[103X] -
  alp[[3Xn[103X-1] ][122X, where [22Xalp =[122X[10XAlpha( [3XG[103X[10X )[110X (see [2XAlpha[102X ([14X75.2-1[114X)).[133X
  
  
  [1X75.2-3 [33X[0;0YIsBergerCondition[133X[101X
  
  [33X[1;0Y[29X[2XIsBergerCondition[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsBergerCondition[102X( [3Xchi[103X ) [32X property[133X
  
  [33X[0;0YCalled  with  an  irreducible  character [3Xchi[103X of a group [22XG[122X, [2XIsBergerCondition[102X
  returns  [9Xtrue[109X  if [3Xchi[103X satisfies [22XM' ≤ ker(χ)[122X for every normal subgroup [22XM[122X of [22XG[122X
  with the property that [22XM ≤ ker(ψ)[122X holds for all [22Xψ ∈ Irr(G)[122X with [22Xψ(1) < χ(1)[122X,
  and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YCalled  with  a  group  [3XG[103X, [2XIsBergerCondition[102X returns [9Xtrue[109X if all irreducible
  characters of [3XG[103X satisfy the inequality above, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YFor  groups  of  odd  order  the  result  is  always  [9Xtrue[109X  by  a theorem of
  T. R. Berger (see [Ber76, Thm. 2.2]).[133X
  
  [33X[0;0YIn  the  case  that  [9Xfalse[109X  is returned, [2XInfoMonomial[102X ([14X75.1-1[114X) tells about a
  degree for which the inequality is violated.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAlpha( Sl23 );[127X[104X
    [4X[28X[ 1, 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27XAlpha( S4 );[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XDelta( Sl23 );[127X[104X
    [4X[28X[ 1, 2, 0 ][128X[104X
    [4X[25Xgap>[125X [27XDelta( S4 );[127X[104X
    [4X[28X[ 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XIsBergerCondition( S4 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsBergerCondition( Sl23 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XList( Irr( Sl23 ), IsBergerCondition );[127X[104X
    [4X[28X[ true, true, true, false, false, false, true ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( Sl23 ), Degree );[127X[104X
    [4X[28X[ 1, 1, 1, 2, 2, 2, 3 ][128X[104X
  [4X[32X[104X
  
  
  [1X75.3 [33X[0;0YPrimitivity of Characters[133X[101X
  
  [1X75.3-1 TestHomogeneous[101X
  
  [33X[1;0Y[29X[2XTestHomogeneous[102X( [3Xchi[103X, [3XN[103X ) [32X function[133X
  
  [33X[0;0YFor  a  group  character  [3Xchi[103X  of  a  group  [22XG[122X and a normal subgroup [3XN[103X of [22XG[122X,
  [2XTestHomogeneous[102X returns a record with information whether the restriction of
  [3Xchi[103X to [3XN[103X is homogeneous, i.e., is a multiple of an irreducible character.[133X
  
  [33X[0;0Y[3XN[103X  may  be  given  also  as  list  of  conjugacy  class positions w.r.t. the
  character table of [22XG[122X.[133X
  
  [33X[0;0YThe components of the result are[133X
  
  [8X[10XisHomogeneous[110X[8X[108X
        [33X[0;6Y[9Xtrue[109X or [9Xfalse[109X,[133X
  
  [8X[10Xcomment[110X[8X[108X
        [33X[0;6Ya  string  telling  a  reason  for  the  value  of  the  [10XisHomogeneous[110X
        component,[133X
  
  [8X[10Xcharacter[110X[8X[108X
        [33X[0;6Yirreducible   constituent  of  the  restriction,  only  bound  if  the
        restriction had to be checked,[133X
  
  [8X[10Xmultiplicity[110X[8X[108X
        [33X[0;6Ymultiplicity of the [10Xcharacter[110X component in the restriction of [3Xchi[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:= DerivedSubgroup( Sl23 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi:= Irr( Sl23 )[7];[127X[104X
    [4X[28XCharacter( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xtest:= TestHomogeneous( chi, n );;[127X[104X
    [4X[25Xgap>[125X [27Xtest.isHomogeneous;  test.comment;  test.multiplicity;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X"restriction checked"[128X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XDegree( test.character );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xchi:= Irr( Sl23 )[4];[127X[104X
    [4X[28XCharacter( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xcln:= ClassPositionsOfNormalSubgroup( CharacterTable( Sl23 ), n );[127X[104X
    [4X[28X[ 1, 4, 7 ][128X[104X
    [4X[25Xgap>[125X [27XTestHomogeneous( chi, cln );[127X[104X
    [4X[28Xrec( comment := "restricts irreducibly", isHomogeneous := true )[128X[104X
  [4X[32X[104X
  
  [1X75.3-2 IsPrimitiveCharacter[101X
  
  [33X[1;0Y[29X[2XIsPrimitiveCharacter[102X( [3Xchi[103X ) [32X property[133X
  
  [33X[0;0YFor  a  character [3Xchi[103X of a group [22XG[122X, [2XIsPrimitiveCharacter[102X returns [9Xtrue[109X if [3Xchi[103X
  is not induced from any proper subgroup, and [9Xfalse[109X otherwise. This currently
  only works for characters of soluble groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimitiveCharacter( Irr( Sl23 )[4] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPrimitiveCharacter( Irr( Sl23 )[7] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X75.3-3 TestQuasiPrimitive[101X
  
  [33X[1;0Y[29X[2XTestQuasiPrimitive[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsQuasiPrimitive[102X( [3Xchi[103X ) [32X property[133X
  
  [33X[0;0Y[2XTestQuasiPrimitive[102X  returns a record with information about quasiprimitivity
  of  the  group  character  [3Xchi[103X, i.e., whether [3Xchi[103X restricts homogeneously to
  every  normal subgroup of its group. The result record contains at least the
  components [10XisQuasiPrimitive[110X (with value either [9Xtrue[109X or [9Xfalse[109X) and [10Xcomment[110X (a
  string telling a reason for the value of the component [10XisQuasiPrimitive[110X). If
  [3Xchi[103X  is not quasiprimitive then there is additionally a component [10Xcharacter[110X,
  with  value  an  irreducible  constituent of a nonhomogeneous restriction of
  [3Xchi[103X.[133X
  
  [33X[0;0Y[2XIsQuasiPrimitive[102X  returns  [9Xtrue[109X or [9Xfalse[109X, depending on whether the character
  [3Xchi[103X is quasiprimitive.[133X
  
  [33X[0;0YNote  that  for solvable groups, quasiprimitivity is the same as primitivity
  (see [2XIsPrimitiveCharacter[102X ([14X75.3-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xchi:= Irr( Sl23 )[4];[127X[104X
    [4X[28XCharacter( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XTestQuasiPrimitive( chi );[127X[104X
    [4X[28Xrec( comment := "all restrictions checked", isQuasiPrimitive := true )[128X[104X
    [4X[25Xgap>[125X [27Xchi:= Irr( Sl23 )[7];[127X[104X
    [4X[28XCharacter( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xtest:= TestQuasiPrimitive( chi );;[127X[104X
    [4X[25Xgap>[125X [27Xtest.isQuasiPrimitive;  test.comment;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X"restriction checked"[128X[104X
    [4X[25Xgap>[125X [27XDegree( test.character );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X75.3-4 TestInducedFromNormalSubgroup[101X
  
  [33X[1;0Y[29X[2XTestInducedFromNormalSubgroup[102X( [3Xchi[103X[, [3XN[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XIsInducedFromNormalSubgroup[102X( [3Xchi[103X ) [32X property[133X
  
  [33X[0;0Y[2XTestInducedFromNormalSubgroup[102X  returns a record with information whether the
  irreducible  character  [3Xchi[103X  of  the group [22XG[122X is induced from a proper normal
  subgroup  of  [22XG[122X. If the second argument [3XN[103X is present, which must be a normal
  subgroup  of  [22XG[122X or the list of class positions of a normal subgroup of [22XG[122X, it
  is checked whether [3Xchi[103X is induced from [3XN[103X.[133X
  
  [33X[0;0YThe  result  contains always the components [10XisInduced[110X (either [9Xtrue[109X or [9Xfalse[109X)
  and  [10Xcomment[110X  (a  string  telling  a  reason  for the value of the component
  [10XisInduced[110X). In the [9Xtrue[109X case there is a component [10Xcharacter[110X which contains a
  character of a maximal normal subgroup from which [3Xchi[103X is induced.[133X
  
  [33X[0;0Y[2XIsInducedFromNormalSubgroup[102X  returns  [9Xtrue[109X  if  [3Xchi[103X is induced from a proper
  normal subgroup of [22XG[122X, and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( Irr( Sl23 ), IsInducedFromNormalSubgroup );[127X[104X
    [4X[28X[ false, false, false, false, false, false, true ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( S4 ){ [ 1, 3, 4 ] },[127X[104X
    [4X[25X>[125X [27X         TestInducedFromNormalSubgroup );[127X[104X
    [4X[28X[ rec( comment := "linear character", isInduced := false ),[128X[104X
    [4X[28X  rec( character := Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X        [ 1, 1, E(3), E(3)^2 ] ),[128X[104X
    [4X[28X      comment := "induced from component '.character'",[128X[104X
    [4X[28X      isInduced := true ),[128X[104X
    [4X[28X  rec( comment := "all maximal normal subgroups checked",[128X[104X
    [4X[28X      isInduced := false ) ][128X[104X
  [4X[32X[104X
  
  
  [1X75.4 [33X[0;0YTesting Monomiality[133X[101X
  
  [33X[0;0YA  character [22Xχ[122X of a finite group [22XG[122X is called [13Xmonomial[113X if [22Xχ[122X is induced from a
  linear character of a subgroup of [22XG[122X. A finite group [22XG[122X is called [13Xmonomial[113X (or
  [13X[22XM[122X-group[113X) if each ordinary irreducible character of [22XG[122X is monomial.[133X
  
  
  [1X75.4-1 [33X[0;0YTestMonomial[133X[101X
  
  [33X[1;0Y[29X[2XTestMonomial[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTestMonomial[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTestMonomial[102X( [3Xchi[103X, [3Xuselattice[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XTestMonomial[102X( [3XG[103X, [3Xuselattice[103X ) [32X operation[133X
  
  [33X[0;0YCalled  with  a  group  character  [3Xchi[103X  of a group [3XG[103X, [2XTestMonomial[102X returns a
  record  containing information about monomiality of the group [3XG[103X or the group
  character [3Xchi[103X, respectively.[133X
  
  [33X[0;0YIf  [2XTestMonomial[102X  proves  the  character  [3Xchi[103X to be monomial then the result
  contains  components [10XisMonomial[110X (with value [9Xtrue[109X), [10Xcomment[110X (a string telling
  a  reason  for  monomiality),  and  if  it was necessary to compute a linear
  character from which [3Xchi[103X is induced, also a component [10Xcharacter[110X.[133X
  
  [33X[0;0YIf  [2XTestMonomial[102X  proves  [3Xchi[103X  or  [3XG[103X to be nonmonomial then the value of the
  component  [10XisMonomial[110X is [9Xfalse[109X, and in the case of [3XG[103X a nonmonomial character
  is  the value of the component [10Xcharacter[110X if it had been necessary to compute
  it.[133X
  
  [33X[0;0YA  Boolean can be entered as the second argument [3Xuselattice[103X; if the value is
  [9Xtrue[109X then the subgroup lattice of the underlying group is used if necessary,
  if  the  value is [9Xfalse[109X then the subgroup lattice is used only for groups of
  order   at  most  [2XTestMonomialUseLattice[102X  ([14X75.4-2[114X).  The  default  value  of
  [3Xuselattice[103X is [9Xfalse[109X.[133X
  
  [33X[0;0YFor  a group whose lattice must not be used, it may happen that [2XTestMonomial[102X
  cannot  prove  or  disprove monomiality; then the result record contains the
  component  [10XisMonomial[110X  with  value  [10X"?"[110X.  This case occurs in the call for a
  character [3Xchi[103X if and only if [3Xchi[103X is not induced from the inertia subgroup of
  a component of any reducible restriction to a normal subgroup. It can happen
  that  [3Xchi[103X is monomial in this situation. For a group, this case occurs if no
  irreducible character can be proved to be nonmonomial, and if no decision is
  possible for at least one irreducible character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTestMonomial( S4 );[127X[104X
    [4X[28Xrec( comment := "abelian by supersolvable group", isMonomial := true )[128X[104X
    [4X[25Xgap>[125X [27XTestMonomial( Sl23 );[127X[104X
    [4X[28Xrec( comment := "list Delta( G ) contains entry > 1",[128X[104X
    [4X[28X  isMonomial := false )[128X[104X
  [4X[32X[104X
  
  [1X75.4-2 TestMonomialUseLattice[101X
  
  [33X[1;0Y[29X[2XTestMonomialUseLattice[102X [32X global variable[133X
  
  [33X[0;0YThis  global  variable  controls for which groups the operation [2XTestMonomial[102X
  ([14X75.4-1[114X)  may  compute  the  subgroup  lattice.  The  value  can be set to a
  positive integer or [2Xinfinity[102X ([14X18.2-1[114X), the default is [22X1000[122X.[133X
  
  [1X75.4-3 IsMonomialNumber[101X
  
  [33X[1;0Y[29X[2XIsMonomialNumber[102X( [3Xn[103X ) [32X property[133X
  
  [33X[0;0YFor  a  positive  integer [3Xn[103X, [2XIsMonomialNumber[102X returns [9Xtrue[109X if every solvable
  group  of  order  [3Xn[103X  is  monomial,  and  [9Xfalse[109X  otherwise.  One can also use
  [10XIsMonomial[110X instead.[133X
  
  [33X[0;0YLet  [22Xν_p(n)[122X  denote  the  multiplicity  of  the  prime [22Xp[122X as factor of [22Xn[122X, and
  [22Xord(p,q)[122X the multiplicative order of [22Xp mod q[122X.[133X
  
  [33X[0;0YThen there exists a solvable nonmonomial group of order [22Xn[122X if and only if one
  of the following conditions is satisfied.[133X
  
  [8X1.[108X
        [33X[0;6Y[22Xν_2(n) ≥ 2[122X and there is a [22Xp[122X such that [22Xν_p(n) ≥ 3[122X and [22Xp ≡ -1 mod 4[122X,[133X
  
  [8X2.[108X
        [33X[0;6Y[22Xν_2(n) ≥ 3[122X and there is a [22Xp[122X such that [22Xν_p(n) ≥ 3[122X and [22Xp ≡ 1 mod 4[122X,[133X
  
  [8X3.[108X
        [33X[0;6Ythere  are  odd prime divisors [22Xp[122X and [22Xq[122X of [22Xn[122X such that [22Xord(p,q)[122X is even
        and [22Xord(p,q) < ν_p(n)[122X (especially [22Xν_p(n) ≥ 3[122X),[133X
  
  [8X4.[108X
        [33X[0;6Ythere  is  a  prime  divisor  [22Xq[122X of [22Xn[122X such that [22Xν_2(n) ≥ 2 ord(2,q) + 2[122X
        (especially [22Xν_2(n) ≥ 4[122X),[133X
  
  [8X5.[108X
        [33X[0;6Y[22Xν_2(n)  ≥  2[122X  and there is a [22Xp[122X such that [22Xp ≡ 1 mod 4[122X, [22Xord(p,q)[122X is odd,
        and [22X2 ord(p,q) < ν_p(n)[122X (especially [22Xν_p(n) ≥ 3[122X).[133X
  
  [33X[0;0YThese  five  possibilities  correspond to the five types of solvable minimal
  nonmonomial  groups (see [2XMinimalNonmonomialGroup[102X ([14X75.5-2[114X)) that can occur as
  subgroups and factor groups of groups of order [3Xn[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFiltered( [ 1 .. 111 ], x -> not IsMonomial( x ) );[127X[104X
    [4X[28X[ 24, 48, 72, 96, 108 ][128X[104X
  [4X[32X[104X
  
  
  [1X75.4-4 [33X[0;0YTestMonomialQuick[133X[101X
  
  [33X[1;0Y[29X[2XTestMonomialQuick[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTestMonomialQuick[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XTestMonomialQuick[102X  does  some  cheap tests whether the irreducible character
  [3Xchi[103X  or  the  group  [3XG[103X,  respectively,  is  monomial.  Here  [21Xcheap[121X  means in
  particular  that no computations of character tables are involved, and it is
  [13Xnot[113X  checked whether [3Xchi[103X is a character and irreducible. The return value is
  a record with components[133X
  
  [8X[10XisMonomial[110X[8X[108X
        [33X[0;6Yeither  [9Xtrue[109X  or  [9Xfalse[109X  or  the  string  [10X"?"[110X,  depending  on  whether
        (non)monomiality could be proved, and[133X
  
  [8X[10Xcomment[110X[8X[108X
        [33X[0;6Ya string telling the reason for the value of the [10XisMonomial[110X component.[133X
  
  [33X[0;0YA group [3XG[103X is proved to be monomial by [2XTestMonomialQuick[102X if [3XG[103X is nilpotent or
  Sylow  abelian  by  supersolvable,  or if [3XG[103X is solvable and its order is not
  divisible by the third power of a prime, Nonsolvable groups are proved to be
  nonmonomial by [2XTestMonomialQuick[102X.[133X
  
  [33X[0;0YAn irreducible character [3Xchi[103X is proved to be monomial if it is linear, or if
  its  codegree  is a prime power, or if its group knows to be monomial, or if
  the  factor  group  modulo  the  kernel  can  be  proved  to  be monomial by
  [2XTestMonomialQuick[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTestMonomialQuick( Irr( S4 )[3] );[127X[104X
    [4X[28Xrec( comment := "whole group is monomial", isMonomial := true )[128X[104X
    [4X[25Xgap>[125X [27XTestMonomialQuick( S4 );[127X[104X
    [4X[28Xrec( comment := "abelian by supersolvable group", isMonomial := true )[128X[104X
    [4X[25Xgap>[125X [27XTestMonomialQuick( Sl23 );[127X[104X
    [4X[28Xrec( comment := "no decision by cheap tests", isMonomial := "?" )[128X[104X
  [4X[32X[104X
  
  
  [1X75.4-5 [33X[0;0YTestSubnormallyMonomial[133X[101X
  
  [33X[1;0Y[29X[2XTestSubnormallyMonomial[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTestSubnormallyMonomial[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsSubnormallyMonomial[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsSubnormallyMonomial[102X( [3Xchi[103X ) [32X property[133X
  
  [33X[0;0YAn  irreducible  character of the group [22XG[122X is called [13Xsubnormally monomial[113X ([13XSM[113X
  for  short) if it is induced from a linear character of a subnormal subgroup
  of [22XG[122X. A group [22XG[122X is called SM if all its irreducible characters are SM.[133X
  
  [33X[0;0Y[2XTestSubnormallyMonomial[102X  returns a record with information whether the group
  [3XG[103X or the irreducible character [3Xchi[103X of [3XG[103X is SM.[133X
  
  [33X[0;0YThe  result  has the components [10XisSubnormallyMonomial[110X (either [9Xtrue[109X or [9Xfalse[109X)
  and  [10Xcomment[110X  (a  string  telling  a  reason  for the value of the component
  [10XisSubnormallyMonomial[110X); in the case that the [10XisSubnormallyMonomial[110X component
  has  value  [9Xfalse[109X  there  is  also  a  component  [10Xcharacter[110X,  with  value an
  irreducible character of [22XG[122X that is not SM.[133X
  
  [33X[0;0Y[2XIsSubnormallyMonomial[102X returns [9Xtrue[109X if the group [3XG[103X or the group character [3Xchi[103X
  is subnormally monomial, and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTestSubnormallyMonomial( S4 );[127X[104X
    [4X[28Xrec( character := Character( CharacterTable( S4 ), [ 3, -1, -1, 0, 1[128X[104X
    [4X[28X     ] ), comment := "found non-SM character",[128X[104X
    [4X[28X  isSubnormallyMonomial := false )[128X[104X
    [4X[25Xgap>[125X [27XTestSubnormallyMonomial( Irr( S4 )[4] );[127X[104X
    [4X[28Xrec( comment := "all subnormal subgroups checked",[128X[104X
    [4X[28X  isSubnormallyMonomial := false )[128X[104X
    [4X[25Xgap>[125X [27XTestSubnormallyMonomial( DerivedSubgroup( S4 ) );[127X[104X
    [4X[28Xrec( comment := "all irreducibles checked",[128X[104X
    [4X[28X  isSubnormallyMonomial := true )[128X[104X
  [4X[32X[104X
  
  
  [1X75.4-6 [33X[0;0YTestRelativelySM[133X[101X
  
  [33X[1;0Y[29X[2XTestRelativelySM[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTestRelativelySM[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTestRelativelySM[102X( [3XG[103X, [3XN[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XTestRelativelySM[102X( [3Xchi[103X, [3XN[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsRelativelySM[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsRelativelySM[102X( [3Xchi[103X ) [32X property[133X
  
  [33X[0;0YIn  the  first two cases, [2XTestRelativelySM[102X returns a record with information
  whether the argument, which must be a SM group [3XG[103X or an irreducible character
  [3Xchi[103X  of a SM group [22XG[122X, is relatively SM with respect to every normal subgroup
  of [3XG[103X.[133X
  
  [33X[0;0YIn  the  second  two cases, a normal subgroup [3XN[103X of [3XG[103X is the second argument.
  Here  [2XTestRelativelySM[102X  returns  a record with information whether the first
  argument  is  relatively  SM  with  respect  to  [3XN[103X,  i.e, whether there is a
  subnormal  subgroup  [22XH[122X  of  [22XG[122X  that  contains  [3XN[103X such that the character [3Xchi[103X
  resp. every  irreducible  character  of [22XG[122X is induced from a character [22Xψ[122X of [22XH[122X
  such that the restriction of [22Xψ[122X to [3XN[103X is irreducible.[133X
  
  [33X[0;0YThe  result record has the components [10XisRelativelySM[110X (with value either [9Xtrue[109X
  or [9Xfalse[109X) and [10Xcomment[110X (a string that describes a reason). If the argument is
  a  group  [3XG[103X that is not relatively SM with respect to a normal subgroup then
  additionally  the  component [10Xcharacter[110X is bound, with value a not relatively
  SM character of such a normal subgroup.[133X
  
  [33X[0;0Y[2XIsRelativelySM[102X  returns  [9Xtrue[109X if the SM group [3XG[103X or the irreducible character
  [3Xchi[103X of the SM group [3XG[103X is relatively SM with respect to every normal subgroup
  of [3XG[103X, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0Y[13XNote[113X that it is [13Xnot[113X checked whether [3XG[103X is SM.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsSubnormallyMonomial( DerivedSubgroup( S4 ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XTestRelativelySM( DerivedSubgroup( S4 ) );[127X[104X
    [4X[28Xrec([128X[104X
    [4X[28X  comment := "normal subgroups are abelian or have nilpotent factor gr\[128X[104X
    [4X[28Xoup", isRelativelySM := true )[128X[104X
  [4X[32X[104X
  
  
  [1X75.5 [33X[0;0YMinimal Nonmonomial Groups[133X[101X
  
  [1X75.5-1 IsMinimalNonmonomial[101X
  
  [33X[1;0Y[29X[2XIsMinimalNonmonomial[102X( [3XG[103X ) [32X property[133X
  
  [33X[0;0YA group [3XG[103X is called [13Xminimal nonmonomial[113X if it is nonmonomial, and all proper
  subgroups and factor groups are monomial.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsMinimalNonmonomial( Sl23 );  IsMinimalNonmonomial( S4 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X75.5-2 MinimalNonmonomialGroup[101X
  
  [33X[1;0Y[29X[2XMinimalNonmonomialGroup[102X( [3Xp[103X, [3Xfactsize[103X ) [32X function[133X
  
  [33X[0;0Yis a solvable minimal nonmonomial group described by the parameters [3Xfactsize[103X
  and [3Xp[103X if such a group exists, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YSuppose  that  the required group [22XK[122X exists. Then [3Xfactsize[103X is the size of the
  Fitting  factor [22XK / F(K)[122X, and this value is 4, 8, an odd prime, twice an odd
  prime, or four times an odd prime. In the case that [3Xfactsize[103X is twice an odd
  prime,  the  centre [22XZ(K)[122X is cyclic of order [22X2^{[3Xp[103X+1}[122X. In all other cases [3Xp[103X is
  the (unique) prime that divides the order of [22XF(K)[122X.[133X
  
  [33X[0;0YThe  solvable  minimal  nonmonomial groups were classified by van der Waall,
  see [{van}76].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalNonmonomialGroup(  2,  3 ); # the group SL(2,3)[127X[104X
    [4X[28X2^(1+2):3[128X[104X
    [4X[25Xgap>[125X [27XMinimalNonmonomialGroup(  3,  4 );[127X[104X
    [4X[28X3^(1+2):4[128X[104X
    [4X[25Xgap>[125X [27XMinimalNonmonomialGroup(  5,  8 );[127X[104X
    [4X[28X5^(1+2):Q8[128X[104X
    [4X[25Xgap>[125X [27XMinimalNonmonomialGroup( 13, 12 );[127X[104X
    [4X[28X13^(1+2):2.D6[128X[104X
    [4X[25Xgap>[125X [27XMinimalNonmonomialGroup(  1, 14 );[127X[104X
    [4X[28X2^(1+6):D14[128X[104X
    [4X[25Xgap>[125X [27XMinimalNonmonomialGroup(  2, 14 );[127X[104X
    [4X[28X(2^(1+6)Y4):D14[128X[104X
  [4X[32X[104X
  
