54.10 TestMonomialQuick

TestMonomialQuick( chi )
TestMonomialQuick( G )

does some easy checks whether the irreducible character chi or the group G are monomial. TestMonomialQuick returns 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 its order is not divisible by the third power of a prime, or if G is nilpotent or Sylow abelian by supersolvable. Nonsolvable groups are proved to me nonmonomial by TestMonomialQuick.

An irreducible character 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 )[3] );
    rec(
      isMonomial := true,
      comment := "kernel factor group is supersolvable" )
    gap> TestMonomialQuick( S4 );
    rec(
      isMonomial := true,
      comment := "abelian by supersolvable group" )
    gap> TestMonomialQuick( Sl23 );
    rec(
      isMonomial := "?",
      comment := "no decision by cheap tests" ) 

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