54.8 TestInducedFromNormalSubgroup

TestInducedFromNormalSubgroup( chi, N )
TestInducedFromNormalSubgroup( chi )

returns a record with information about whether the irreducible character chi of the group G is induced from a proper normal subgroup of G.

If chi is the only argument then it is checked whether there is a maximal normal subgroup of G from that chi is induced. If there is a second argument N, a normal subgroup of G, then it is checked whether chi is induced from N. N may also be given as the list of positions of conjugacy classes contained in the normal subgroup in question.

The result contains the components

isInduced:

true or false,

comment:

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

character:

if bound, a character of a maximal normal subgroup of G or of the argument N from that chi is induced.

IsInducedFromNormalSubgroup( chi )

returns true if the group character chi is induced from a proper normal subgroup of the group of chi, and false otherwise.

    gap> List( Irr( Sl23 ), IsInducedFromNormalSubgroup );
    [ false, false, false, false, false, false, true ]
    gap> List( Irr( S4 ){ [ 1, 3, 4 ] },
    >          TestInducedFromNormalSubgroup );
    #W  Warning: Group has no name
    [ rec(
          isInduced := false,
          comment := "linear character" ), rec(
          isInduced := true,
          comment := "induced from component '.character'",
          character := Character( Subgroup( S4, [ b, c, d ] ), 
            [ 1, 1, E(3), E(3)^2 ] ) ), rec(
          isInduced := false,
          comment := "all maximal normal subgroups checked" ) ] 

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