36.3 Subgroups

For the general concept of parents and subdomains, see Parents and Constructing Subdomains. More functions that construct certain subgroups can be found in the sections Normal Structure, Specific and Parametrized Subgroups, Sylow Subgroups and Hall Subgroups, and Subgroups characterized by prime powers.

  • Subgroup( G, gens ) F
  • SubgroupNC( G, gens ) F

    creates the subgroup U of G generated by gens. The Parent of U will be G. The NC version does not check, whether the elements in gens actually lie in G. 

    gap> u:=Subgroup(g,[(1,2,3),(1,2)]);
Group( [ (1,2,3), (1,2) ] )

  • Index( G, U ) O

    returns the index [G:U] = [(|G|)/( |U|)]. 

    gap> Index(g,u);
4

  • IndexInWholeGroup( G ) A

    If the family of elements of G itself forms a group P, this attribute returns the index of G in P.

  • AsSubgroup( G, U ) O

    creates a subgroup of G which contains the same elements as U 

    gap> v:=AsSubgroup(g,Group((1,2,3),(1,4)));
Group( [ (1,2,3), (1,4) ] )
gap> Parent(v);
Group( [ (1,2,3,4), (1,2) ] )

  • IsSubgroup( G, U ) F

    IsSubgroup returns true if U is a group that is a subset of the domain G. This is actually checked by calling IsGroup( U ) and IsSubset( G, U ); note that special methods for IsSubset (see IsSubset) are available that test only generators of U if G is closed under the group operations. So in most cases, for example whenever one knows already that U is a group, it is better to call only IsSubset. 

    gap> IsSubgroup(g,u);
true
gap> v:=Group((1,2,3),(1,2));    
Group( [ (1,2,3), (1,2) ] )
gap> u=v; 
true
gap> IsSubgroup(g,v);
true

  • IsNormal( G, U ) O

    returns true if the group G normalizes the group U and false otherwise.

    A group G normalizes a group U if and only if for every g Î G and u Î U the element ug is a member of U. Note that U need not be a subgroup of G. 

    gap> IsNormal(g,u);
false

  • IsCharacteristicSubgroup( G, N ) O

    tests whether N is invariant under all automorphisms of G. 

    gap> IsCharacteristicSubgroup(g,u);
false

  • ConjugateSubgroup( G, g ) O

  • ConjugateSubgroups( G, U ) O

    returns a list of all images of the group U under conjugation action by G.

  • IsSubnormal( G, U ) O

    A subgroup U of the group G is subnormal if it is contained in a subnormal series of G. 

    gap> IsSubnormal(g,Group((1,2,3)));   
false
gap> IsSubnormal(g,Group((1,2)(3,4)));
true

    If a group U is created as a subgroup of another group G, G becomes the parent of U. There is no universal parent group, parent-child chains can be arbitrary long. GAP stores the result of some operations (such as Normalizer) with the parent as an attribute.

    [Top] [Previous] [Up] [Next] [Index]

    GAP 4 manual
    February 2000