25.81 ExtendedIntersectionSumAgGroup

ExtendedIntersectionSumAgGroup( V, W )

Let V and W be ag groups with a common parent group, such that <W> leq N(<V>). Then <V> * <W> is a subgroup and ExtendedIntersectionSumAgGroup returns the intersection and the sum of V and W. The information about these groups is returned as a record with the components intersection, sum and the additional information leftSide and rightSide.

intersection:

is bound to the intersection <W> cap <V>.

sum:

is bound to the sum <V> * <W>.

leftSide:

is lists of ag words, see below.

rightSide:

is lists of agwords, see below.

The function uses the Zassenhaus sum-intersection algorithm. Let V be generated by v_1, ..., v_a, W be generated by w_1, ..., w_b. Then the matrix

left( beginarraycc v_1 & 1
vdots & vdots
v_a & 1
w_1 & w_1
vdots & vdots
w_b & w_b
endarray right)

is echelonized by using the sifting algorithm to produce the following matrix

left( beginarraycc l_1 & k_1
vdots & vdots
l_c & k_c
1 & k_c+1
vdots & vdots
1 & k_a+b
endarray right).

Then l_1, ..., l_c is a generating sequence for the sum, while the sequence k_{c+1}, ..., k_{a+b} is is a generating sequence for the intersection. leftSide is bound to a list, such that the i.th list element is l_j, if there exists a j, such that l_j has depth i, and IdAgWord otherwise. rightSide is bound to a list, such that the i.th list element is k_j, if there exists a j less than c+1, such that k_j has depth i, and IdAgWord otherwise. See also SumFactorizationFunctionAgGroup.

Note that this functions returns an incorrect result if <W> not leq N(<V>).

    gap> v4_1 := AgSubgroup( s4, [ a*b, c ], true );
    Subgroup( s4, [ a*b, c ] )
    gap> v4_2 := AgSubgroup( s4, [ c, d ], true );
    Subgroup( s4, [ c, d ] )
    gap> ExtendedIntersectionSumAgGroup( v4_1, v4_2 );
    rec(
      leftSide := [ a*b, IdAgWord, c, d ],
      rightSide := [ IdAgWord, IdAgWord, c, d ],
      sum := Subgroup( s4, [ a*b, c, d ] ),
      intersection := Subgroup( s4, [ c ] ) ) 

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