7.20 ConjugateSubgroup

ConjugateSubgroup( U, g )

ConjugateSubgroup returns the subgroup <U>^<g> conjugate to U under g, which must be an element of the parent group of G.

If present, the flags U.isAbelian, U.isCyclic, U.isElementaryAbelian, U.isFinite, U.isNilpotent, U.isPerfect, U.isSimple, U.isSolvable, and U.size are copied to <U>^<g>.

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

The default function GroupOps.ConjugateSubgroup returns U if the set of elements of U is known and g is an element of this set or if g is a generator of U. Otherwise it conjugates the generators of U with g.

If the set of elements of U is known the default function also conjugates and binds it to the conjugate subgroup.

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