7.16 Centralizer

Centralizer( G, x )

Centralizer returns the centralizer of an element x in G where x must be an element of the parent group of G.

The centralizer of an element x in G is defined as the set C of elements c of G such that c and x commute.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> v4 := Centralizer( s4, (1,2) );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (3,4), (1,2) ] )

The default function GroupOps.Centralizer uses Stabilizer (see Stabilizer) in order to compute the centralizer of x in G acting by conjugation.

Centralizer( G, U )

Centralizer returns the centralizer of a group U in G as group record. Note that G and U must have a common parent group.

The centralizer of a group U in G is defined as the set C of elements c of C such c commutes with every element of U.

If G is the parent group of U then Centralizer will set and test the record component U.centralizer.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> v4 := Centralizer( s4, (1,2) );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (3,4), (1,2) ] )
    gap> c2 := Subgroup( s4, [ (1,3) ] );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (1,3) ] )
    gap> Centralizer( v4, c2 );
    Subgroup( Group( (1,2,3,4), (1,2) ), [  ] ) 

The default function GroupOps.Centralizer uses Stabilizer in order to compute successively the stabilizer of the generators of U.

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