GAP Manual: 7.18. Closure

Closure( U, g )

Let U be a group with parent group G and let g be an element of G. Then Closure returns the closure C of U and g as subgroup of G. The closure C of U and g is the subgroup generated by U and g. 

    gap> s4 := Group( (1,2,3,4), (1,2 ) );
    Group( (1,2,3,4), (1,2) )
    gap> s2 := Subgroup( s4, [ (1,2) ] );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2) ] )
    gap> Closure( s2, (3,4) );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2), (3,4) ] )

The default function GroupOps.Closure returns U if U is a parent group, or if g or its inverse is a generator of U, or if the set of elements is known and g is in this set, or if g is trivial. Otherwise the function constructs a new subgroup C which is generated by the generators of U and the element g.

Note that if the set of elements of U is bound to U.elements then GroupOps.Closure computes the set of elements for C and binds it to C.elements.

If U is known to be non-abelian or infinite so is C. If U is known to be abelian the function checks whether g commutes with every generator of U.

Closure( U, S )

Let U and S be two group with a common parent group G. Then Closure returns the subgroup of G generated by U and S. 

    gap> s4 := Group( (1,2,3,4), (1,2 ) );
    Group( (1,2,3,4), (1,2) )
    gap> s2 := Subgroup( s4, [ (1,2) ] );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2) ] )
    gap> z3 := Subgroup( s4, [ (1,2,3) ] );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2,3) ] )
    gap> Closure( z3, s2 );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,2,3), (1,2) ] )

The default function GroupOps.Closure returns the parent of U and S if U or S is a parent group. Otherwise the function computes the closure of U under all generators of S.

Note that if the set of elements of U is bound to U.elements then GroupOps.Closure computes the set of elements for the closure C and binds it to C.elements.

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