51.16 ConsiderKernels

ConsiderKernels( tbl, chars, prime_powermap, prime )
ConsiderKernels( tbl, chars, prime_powermap, prime, "quick" )

More about Maps and Parametrized Maps) that is an approximation of the prime-th powermap of the character table tbl:

For G a group with character table tbl, the kernel of each character in the list chars is a normal subgroup of G, so for every g in 'Kernel( <chi> )' we have g^{<prime>} in 'Kernel( <chi> )'.

Depending on the order of the factor group modulo Kernel( chi ), there are two further properties: If the order is relative prime to prime, for each g notin 'Kernel( <chi> )' the prime-th power is not contained in Kernel( chi ); if the order is equal to prime, the prime-th powers of all elements lie in Kernel( chi ).

If "quick" is specified, only those classes are considered for which prime_powermap is ambiguous.

If Kernel( chi ) has an order not dividing tbl.order for an element chi of chars, or if no image is possible for a class, false is returned; otherwise ConsiderKernels returns true.

Note that chars must consist of ordinary characters, since the kernel of a virtual character is not defined.

    # see example in "Congruences"
    gap> ConsiderKernels( t, t.irreducibles, pow, 2 ); pow;
    true
    [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, [ 6, 7 ], 14, 9, 1, 1, 2, 2,
      3, 4, 5, [ 6, 7, 18, 19 ], 8, 9, 9, [ 10, 21 ], 11, 12, 16, 16,
      [ 2, 16 ], [ 2, 16 ], 17, 17, [ 6, 18 ], [ 6, 18 ], 
      [ 6, 7, 18, 19 ], [ 6, 7, 18, 19 ], 20, 20, [ 9, 20 ], [ 9, 20 ], 
      22, 22, 24, 24, [ 15, 25, 26 ], [ 15, 25, 26 ], 28, 28, 29, 29 ]
    # continued in "PowermapsAllowedBySymmetrisations"

ConsiderKernels is used by Powermap Powermap.

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