7.21 Core

Core( S, U )

Let S and U be groups with a common parent group G. Then Core returns the core of U under conjugation of S.

The core of a group U under a group S Core_{<S>}( <U> ) is the intersection bigcap_{s in <S>} <U>^s of all groups conjugate to U under conjugation by elements of S.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> s4.name := "s4";;
    gap> d8 := Subgroup( s4, [ (1,2,3,4), (1,2)(3,4) ] );
    Subgroup( s4, [ (1,2,3,4), (1,2)(3,4) ] )
    gap> Core( s4, d8 );
    Subgroup( s4, [ (1,2)(3,4), (1,3)(2,4) ] )
    gap> Core( d8, s4 );
    s4 

The default function GroupOps.Core starts with U and replaces U with the intersection of U and a conjugate subgroup of U under a generator of G until the subgroup is normalized by G.

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