54.11 TestMonomial

TestMonomial( chi )
TestMonomial( G )

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

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

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

If the program 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.

IsMonomial( G ) IsMonomial( chi )

returns true if the group G or the character chi of a solvable group can be proved to be monomial, false if it can be proved to be nonmonomial, and the string "?" otherwise.

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

IsMonomial( n )

for a positive integer n returns true if every solvable group of order n is monomial, and false otherwise.

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

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