### 75 Monomiality Questions

This chapter describes functions dealing with the monomiality of finite (solvable) groups and their characters.

All these functions assume characters to be class function objects as described in Chapter 72, lists of character values are not allowed.

The usual property tests of GAP that return either true or false are not sufficient for us. When we ask whether a group character $$\chi$$ has a certain property, such as quasiprimitivity, we usually want more information than just yes or no. Often we are interested in the reason why a group character $$\chi$$ was proved to have a certain property, e.g., whether monomiality of $$\chi$$ 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 $$\chi$$ 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 TestQuasiPrimitive (75.3-3) contains the component isQuasiPrimitive (which is the known boolean property flag), and additionally the component comment, a string telling the reason for the value of the isQuasiPrimitive component, and in the case that the argument $$\chi$$ was not quasiprimitive also the component character, which is an irreducible constituent of a nonhomogeneous restriction of $$\chi$$ to a normal subgroup. Besides these test functions there are also the known properties, e.g., the property IsQuasiPrimitive (75.3-3) which will call the attribute TestQuasiPrimitive (75.3-3), and return the value of the isQuasiPrimitive component of the result.

A 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 $$N$$ that was constructed as a normal closure of some group elements, and suppose that you have already computed with normal subgroups (by calls to NormalSubgroups (39.19-9) or MaximalNormalSubgroups (39.19-10)) and their character tables. Then you should look in the lists of known normal subgroups whether $$N$$ is contained, and if so you can use the known character table. A mechanism that supports this for normal subgroups is described in 71.23.

Also 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 $$G$$ then it may be advisable to compute it as soon as possible. For example, if you need the normal subgroups of $$G$$ then they can be computed more efficiently if the character table of $$G$$ is known, and they can be stored compatibly to the contained $$G$$-conjugacy classes. This correspondence of classes list and normal subgroup can be used very often.

Several examples in this chapter use the symmetric group $$S_4$$ and the special linear group $$SL(2,3)$$. For running the examples, you must first define the groups, for example as follows.

gap> S4:= SymmetricGroup( 4 );;  SetName( S4, "S4" );
gap> Sl23:= SL( 2, 3 );;


#### 75.1 InfoMonomial (Info Class)

##### 75.1-1 InfoMonomial
 ‣ InfoMonomial ( info class )

Most of the functions described in this chapter print some (hopefully useful) information if the info level of the info class InfoMonomial is at least $$1$$, see 7.4 for details.

#### 75.2 Character Degrees and Derived Length

##### 75.2-1 Alpha
 ‣ Alpha( G ) ( attribute )

For a group G, Alpha returns a list whose $$i$$-th entry is the maximal derived length of groups $$\textit{G} / \ker(\chi)$$ for $$\chi \in Irr(\textit{G})$$ with $$\chi(1)$$ at most the $$i$$-th irreducible degree of G.

##### 75.2-2 Delta
 ‣ Delta( G ) ( attribute )

For a group G, Delta returns the list $$[ 1, alp - alp, \ldots, alp[\textit{n}] - alp[\textit{n}-1] ]$$, where $$alp =$$Alpha( G ) (see Alpha (75.2-1)).

##### 75.2-3 IsBergerCondition
 ‣ IsBergerCondition( G ) ( property )
 ‣ IsBergerCondition( chi ) ( property )

Called with an irreducible character chi of a group $$G$$, IsBergerCondition returns true if chi satisfies $$M' \leq \ker(\chi)$$ for every normal subgroup $$M$$ of $$G$$ with the property that $$M \leq \ker(\psi)$$ holds for all $$\psi \in Irr(G)$$ with $$\psi(1) < \chi(1)$$, and false otherwise.

Called with a group G, IsBergerCondition returns true if all irreducible characters of G satisfy the inequality above, and false otherwise.

For groups of odd order the result is always true by a theorem of T. R. Berger (see [Ber76, Thm. 2.2]).

In the case that false is returned, InfoMonomial (75.1-1) tells about a degree for which the inequality is violated.

gap> Alpha( Sl23 );
[ 1, 3, 3 ]
gap> Alpha( S4 );
[ 1, 2, 3 ]
gap> Delta( Sl23 );
[ 1, 2, 0 ]
gap> Delta( S4 );
[ 1, 1, 1 ]
gap> IsBergerCondition( S4 );
true
gap> IsBergerCondition( Sl23 );
false
gap> List( Irr( Sl23 ), IsBergerCondition );
[ true, true, true, false, false, false, true ]
gap> List( Irr( Sl23 ), Degree );
[ 1, 1, 1, 2, 2, 2, 3 ]


#### 75.3 Primitivity of Characters

##### 75.3-1 TestHomogeneous
 ‣ TestHomogeneous( chi, N ) ( function )

For a group character chi of the group $$G$$, say, and a normal subgroup N of $$G$$, TestHomogeneous returns a record with information whether the restriction of chi to N is homogeneous, i.e., is a multiple of an irreducible character.

N may be given also as list of conjugacy class positions w.r.t. the character table of $$G$$.

The components of the result are

isHomogeneous

true or false,

comment

a string telling a reason for the value of the isHomogeneous component,

character

irreducible constituent of the restriction, only bound if the restriction had to be checked,

multiplicity

multiplicity of the character component in the restriction of chi.

gap> n:= DerivedSubgroup( Sl23 );;
gap> chi:= Irr( Sl23 );
Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] )
gap> TestHomogeneous( chi, n );
rec( character := Character( CharacterTable( Group(
[ [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ],
[ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ] ]) ),
[ 1, -1, 1, -1, 1 ] ), comment := "restriction checked",
isHomogeneous := false, multiplicity := 1 )
gap> chi:= Irr( Sl23 );
Character( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] )
gap> cln:= ClassPositionsOfNormalSubgroup( CharacterTable( Sl23 ), n );
[ 1, 4, 7 ]
gap> TestHomogeneous( chi, cln );
rec( comment := "restricts irreducibly", isHomogeneous := true )


##### 75.3-2 IsPrimitiveCharacter
 ‣ IsPrimitiveCharacter( chi ) ( property )

For a character chi of the group $$G$$, say, IsPrimitiveCharacter returns true if chi is not induced from any proper subgroup, and false otherwise.

gap> IsPrimitive( Irr( Sl23 ) );
true
gap> IsPrimitive( Irr( Sl23 ) );
false


##### 75.3-3 TestQuasiPrimitive
 ‣ TestQuasiPrimitive( chi ) ( attribute )
 ‣ IsQuasiPrimitive( chi ) ( property )

TestQuasiPrimitive returns a record with information about quasiprimitivity of the group character chi, i.e., whether chi restricts homogeneously to every normal subgroup of its group. The result record contains at least the components isQuasiPrimitive (with value either true or false) and comment (a string telling a reason for the value of the component isQuasiPrimitive). If chi is not quasiprimitive then there is additionally a component character, with value an irreducible constituent of a nonhomogeneous restriction of chi.

IsQuasiPrimitive returns true or false, depending on whether the character chi is quasiprimitive.

Note that for solvable groups, quasiprimitivity is the same as primitivity (see IsPrimitiveCharacter (75.3-2)).

gap> chi:= Irr( Sl23 );
Character( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] )
gap> TestQuasiPrimitive( chi );
rec( comment := "all restrictions checked", isQuasiPrimitive := true )
gap> chi:= Irr( Sl23 );
Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] )
gap> TestQuasiPrimitive( chi );
rec( character := Character( CharacterTable( Group(
[ [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ],
[ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ] ]) ),
[ 1, -1, 1, -1, 1 ] ), comment := "restriction checked",
isQuasiPrimitive := false )


##### 75.3-4 TestInducedFromNormalSubgroup
 ‣ TestInducedFromNormalSubgroup( chi[, N] ) ( function )
 ‣ IsInducedFromNormalSubgroup( chi ) ( property )

TestInducedFromNormalSubgroup returns a record with information whether the irreducible character chi of the group $$G$$, say, is induced from a proper normal subgroup of $$G$$. If the second argument N is present, which must be a normal subgroup of $$G$$ or the list of class positions of a normal subgroup of $$G$$, it is checked whether chi is induced from N.

The result contains always the components isInduced (either true or false) and comment (a string telling a reason for the value of the component isInduced). In the true case there is a component character which contains a character of a maximal normal subgroup from which chi is induced.

IsInducedFromNormalSubgroup returns true if chi is induced from a proper normal subgroup of $$G$$, and false otherwise.

gap> List( Irr( Sl23 ), IsInducedFromNormalSubgroup );
[ false, false, false, false, false, false, true ]
gap> List( Irr( S4 ){ [ 1, 3, 4 ] },
>          TestInducedFromNormalSubgroup );
[ rec( comment := "linear character", isInduced := false ),
rec( character := Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
[ 1, 1, E(3)^2, E(3) ] ),
comment := "induced from component '.character'",
isInduced := true ),
rec( comment := "all maximal normal subgroups checked",
isInduced := false ) ]


#### 75.4 Testing Monomiality

A character $$\chi$$ of a finite group $$G$$ is called monomial if $$\chi$$ is induced from a linear character of a subgroup of $$G$$. A finite group $$G$$ is called monomial (or $$M$$-group) if each ordinary irreducible character of $$G$$ is monomial.

##### 75.4-1 TestMonomial
 ‣ TestMonomial( chi ) ( attribute )
 ‣ TestMonomial( G ) ( attribute )
 ‣ TestMonomial( chi, uselattice ) ( operation )
 ‣ TestMonomial( G, uselattice ) ( operation )

Called with a group character chi of a group G, TestMonomial returns a record containing information about monomiality of the group G or the group character chi, respectively.

If TestMonomial proves the character chi to be monomial then the result contains components isMonomial (with value true), comment (a string telling a reason for monomiality), and if it was necessary to compute a linear character from which chi is induced, also a component character.

If TestMonomial proves chi or G to be nonmonomial then the value of the component isMonomial is false, and in the case of G a nonmonomial character is the value of the component character if it had been necessary to compute it.

A Boolean can be entered as the second argument uselattice; if the value is true then the subgroup lattice of the underlying group is used if necessary, if the value is false then the subgroup lattice is used only for groups of order at most TestMonomialUseLattice (75.4-2). The default value of uselattice is false.

For a group whose lattice must not be used, it may happen that TestMonomial cannot prove or disprove monomiality; then the result record contains the component isMonomial with value "?". This case occurs in the call for a character chi if and only if chi is not induced from the inertia subgroup of a component of any reducible restriction to a normal subgroup. It can happen that chi 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.

gap> TestMonomial( S4 );
rec( comment := "abelian by supersolvable group", isMonomial := true )
gap> TestMonomial( Sl23 );
rec( comment := "list Delta( G ) contains entry > 1",
isMonomial := false )


##### 75.4-2 TestMonomialUseLattice
 ‣ TestMonomialUseLattice ( global variable )

This global variable controls for which groups the operation TestMonomial (75.4-1) may compute the subgroup lattice. The value can be set to a positive integer or infinity (18.2-1), the default is $$1000$$.

##### 75.4-3 IsMonomialNumber
 ‣ IsMonomialNumber( n ) ( property )

For a positive integer n, IsMonomialNumber returns true if every solvable group of order n is monomial, and false otherwise. One can also use IsMonomial instead.

Let $$\nu_p(n)$$ denote the multiplicity of the prime $$p$$ as factor of $$n$$, and $$ord(p,q)$$ the multiplicative order of $$p \pmod{q}$$.

Then there exists a solvable nonmonomial group of order $$n$$ if and only if one of the following conditions is satisfied.

1.

$$\nu_2(n) \geq 2$$ and there is a $$p$$ such that $$\nu_p(n) \geq 3$$ and $$p \equiv -1 \pmod{4}$$,

2.

$$\nu_2(n) \geq 3$$ and there is a $$p$$ such that $$\nu_p(n) \geq 3$$ and $$p \equiv 1 \pmod{4}$$,

3.

there are odd prime divisors $$p$$ and $$q$$ of $$n$$ such that $$ord(p,q)$$ is even and $$ord(p,q) < \nu_p(n)$$ (especially $$\nu_p(n) \geq 3$$),

4.

there is a prime divisor $$q$$ of $$n$$ such that $$\nu_2(n) \geq 2 ord(2,q) + 2$$ (especially $$\nu_2(n) \geq 4$$),

5.

$$\nu_2(n) \geq 2$$ and there is a $$p$$ such that $$p \equiv 1 \pmod{4}$$, $$ord(p,q)$$ is odd, and $$2 ord(p,q) < \nu_p(n)$$ (especially $$\nu_p(n) \geq 3$$).

These five possibilities correspond to the five types of solvable minimal nonmonomial groups (see MinimalNonmonomialGroup (75.5-2)) that can occur as subgroups and factor groups of groups of order n.

gap> Filtered( [ 1 .. 111 ], x -> not IsMonomial( x ) );
[ 24, 48, 72, 96, 108 ]


##### 75.4-4 TestMonomialQuick
 ‣ TestMonomialQuick( chi ) ( attribute )
 ‣ TestMonomialQuick( G ) ( attribute )

TestMonomialQuick does some cheap tests whether the irreducible character chi or the group G, respectively, is monomial. Here "cheap" means in particular that no computations of character tables are involved, and it is not checked whether chi is a character and irreducible. The return value is a record with components

isMonomial

either true or false or the string "?", depending on whether (non)monomiality could be proved, and

comment

a string telling the reason for the value of the isMonomial component.

A group G is proved to be monomial by TestMonomialQuick if G is nilpotent or Sylow abelian by supersolvable, or if G is solvable and its order is not divisible by the third power of a prime, Nonsolvable groups are proved to be nonmonomial by TestMonomialQuick.

An irreducible character chi 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 TestMonomialQuick.

gap> TestMonomialQuick( Irr( S4 ) );
rec( comment := "whole group is monomial", isMonomial := true )
gap> TestMonomialQuick( S4 );
rec( comment := "abelian by supersolvable group", isMonomial := true )
gap> TestMonomialQuick( Sl23 );
rec( comment := "no decision by cheap tests", isMonomial := "?" )


##### 75.4-5 TestSubnormallyMonomial
 ‣ TestSubnormallyMonomial( G ) ( attribute )
 ‣ TestSubnormallyMonomial( chi ) ( attribute )
 ‣ IsSubnormallyMonomial( G ) ( property )
 ‣ IsSubnormallyMonomial( chi ) ( property )

An irreducible character of the group $$G$$ is called subnormally monomial (SM for short) if it is induced from a linear character of a subnormal subgroup of $$G$$. A group $$G$$ is called SM if all its irreducible characters are SM.

TestSubnormallyMonomial returns a record with information whether the group G or the irreducible character chi of G is SM.

The result has the components isSubnormallyMonomial (either true or false) and comment (a string telling a reason for the value of the component isSubnormallyMonomial); in the case that the isSubnormallyMonomial component has value false there is also a component character, with value an irreducible character of $$G$$ that is not SM.

IsSubnormallyMonomial returns true if the group G or the group character chi is subnormally monomial, and false otherwise.

gap> TestSubnormallyMonomial( S4 );
rec( character := Character( CharacterTable( S4 ), [ 3, -1, -1, 0, 1
] ), comment := "found non-SM character",
isSubnormallyMonomial := false )
gap> TestSubnormallyMonomial( Irr( S4 ) );
rec( comment := "all subnormal subgroups checked",
isSubnormallyMonomial := false )
gap> TestSubnormallyMonomial( DerivedSubgroup( S4 ) );
rec( comment := "all irreducibles checked",
isSubnormallyMonomial := true )


##### 75.4-6 TestRelativelySM
 ‣ TestRelativelySM( G ) ( attribute )
 ‣ TestRelativelySM( chi ) ( attribute )
 ‣ TestRelativelySM( G, N ) ( operation )
 ‣ TestRelativelySM( chi, N ) ( operation )
 ‣ IsRelativelySM( G ) ( property )
 ‣ IsRelativelySM( chi ) ( property )

In the first two cases, TestRelativelySM returns a record with information whether the argument, which must be a SM group G or an irreducible character chi of a SM group $$G$$, is relatively SM with respect to every normal subgroup of G.

In the second two cases, a normal subgroup N of G is the second argument. Here TestRelativelySM returns a record with information whether the first argument is relatively SM with respect to N, i.e, whether there is a subnormal subgroup $$H$$ of $$G$$ that contains N such that the character chi resp. every irreducible character of $$G$$ is induced from a character $$\psi$$ of $$H$$ such that the restriction of $$\psi$$ to N is irreducible.

The result record has the components isRelativelySM (with value either true or false) and comment (a string that describes a reason). If the argument is a group G that is not relatively SM with respect to a normal subgroup then additionally the component character is bound, with value a not relatively SM character of such a normal subgroup.

IsRelativelySM returns true if the SM group G or the irreducible character chi of the SM group G is relatively SM with respect to every normal subgroup of G, and false otherwise.

Note that it is not checked whether G is SM.

gap> IsSubnormallyMonomial( DerivedSubgroup( S4 ) );
true
gap> TestRelativelySM( DerivedSubgroup( S4 ) );
rec(
comment := "normal subgroups are abelian or have nilpotent factor gr\
oup", isRelativelySM := true )


#### 75.5 Minimal Nonmonomial Groups

##### 75.5-1 IsMinimalNonmonomial
 ‣ IsMinimalNonmonomial( G ) ( property )

A group G is called minimal nonmonomial if it is nonmonomial, and all proper subgroups and factor groups are monomial.

gap> IsMinimalNonmonomial( Sl23 );  IsMinimalNonmonomial( S4 );
true
false


##### 75.5-2 MinimalNonmonomialGroup
 ‣ MinimalNonmonomialGroup( p, factsize ) ( function )

is a solvable minimal nonmonomial group described by the parameters factsize and p if such a group exists, and false otherwise.

Suppose that the required group $$K$$ exists. Then factsize is the size of the Fitting factor $$K / F(K)$$, and this value is 4, 8, an odd prime, twice an odd prime, or four times an odd prime. In the case that factsize is twice an odd prime, the centre $$Z(K)$$ is cyclic of order $$2^{{\textit{p}+1}}$$. In all other cases p is the (unique) prime that divides the order of $$F(K)$$.

The solvable minimal nonmonomial groups were classified by van der Waall, see [vdW76].

gap> MinimalNonmonomialGroup(  2,  3 ); # the group SL(2,3)
2^(1+2):3
gap> MinimalNonmonomialGroup(  3,  4 );
3^(1+2):4
gap> MinimalNonmonomialGroup(  5,  8 );
5^(1+2):Q8
gap> MinimalNonmonomialGroup( 13, 12 );
13^(1+2):2.D6
gap> MinimalNonmonomialGroup(  1, 14 );
2^(1+6):D14
gap> MinimalNonmonomialGroup(  2, 14 );
(2^(1+6)Y4):D14


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