A group homomorphism is a mapping from one group to another that respects
multiplication and inverses. They are implemented as a special class of
mappings, so in particular all operations for mappings, such as
IsSurjective (see chapter Mappings, in particular section Mappings that Respect Multiplication) are applicable to them.
Homomorphisms can be used to transfer calculations into isomorphic groups in another representation, for which better algoroithms are available. Section Nice Monomorphisms explains a technique how to enforce this automatically.
Homomorphisms are also used to represent group automorphisms, and section Group Automorphisms explains explains GAP's facilities to work with automorphism groups.
The penultimate section of this chapter, Searching for Homomorphisms, explains how to make GAP to search for all homomorphisms between two groups which fulfill certain specifications.
37.1 Creating Group Homomorphisms
The most important way of creating group homomorphisms is to give images for a set of group generators and to extend it to the group generated by them by the homomorphism property.
GroupHomomorphismByImages returns the group homomorphism with
source G and range H that is defined by mapping the list gens of
generators of G to the list imgs of images in H.
If gens does not generate G or if the mapping of the generators does
not extend to a homomorphism
(i.e., if mapping the generators describes only a multi-valued mapping)
fail is returned.
This test can be quite expensive. If one is certain that the mapping of
the generators extends to a homomorphism,
one can avoid the checks by calling
(There also is the possibility to
construct potentially multi-valued mappings with
GroupGeneralMappingByImages and to test with
they are indeed homomorphisms.)
GroupHomomorphismByImagesNC creates a homomorphism as
GroupHomomorphismByImages does, however it does not test whether
gens generates G and that the mapping of
gens to imgs indeed defines a group homomorphism.
Because these tests can be expensive it can be substantially faster than
Results are unpredictable if the conditions do not hold.
(For creating a possibly multi-valued mapping from G to H that
respects multiplication and inverses,
GroupGeneralMappingByImages can be used.)
gap> gens:=[(1,2,3,4),(1,2)]; [ (1,2,3,4), (1,2) ] gap> g:=Group(gens); Group([ (1,2,3,4), (1,2) ]) gap> h:=Group((1,2,3),(1,2)); Group([ (1,2,3), (1,2) ]) gap> hom:=GroupHomomorphismByImages(g,h,gens,[(1,2),(1,3)]); [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,3) ] gap> Image(hom,(1,4)); (2,3) gap> map:=GroupHomomorphismByImages(g,h,gens,[(1,2,3),(1,2)]); fail
returns a generalized mapping defined by extending the mapping from
gensG to gensH homomorphically.
GroupHomomorphismByImages creates a
tests whether it
gap> map:=GroupGeneralMappingByImages(g,h,gens,[(1,2,3),(1,2)]); [ (1,2,3,4), (1,2) ] -> [ (1,2,3), (1,2) ] gap> IsMapping(map); false
A second way to create homomorphisms is to give functions that compute image and preimage. (A similar case are homomorphisms that are induced by conjugation. Special constructors for such mappings are described in section Group Automorphisms).
GroupHomomorphismByFunction returns a group homomorphism hom with
source S and range R, such that each element s of S is mapped to
), where fun is a GAP function.
If the argument invfun is bound then hom is a bijection between S
and R, and the preimage of each element r of R is given by
), where invfun is a GAP function.
No test is performed on whether the functions actually give an homomorphism between both groups because this would require testing the full multiplication table.
GroupHomomorphismByFunction creates a mapping which
gap> hom:=GroupHomomorphismByFunction(g,h, > function(x) if SignPerm(x)=-1 then return (1,2); else return ();fi;end); MappingByFunction( Group([ (1,2,3,4), (1,2) ]), Group([ (1,2,3), (1,2) ]), function( x ) ... end ) gap> ImagesSource(hom); Group([ (1,2), (1,2) ]) gap> Image(hom,(1,2,3,4)); (1,2)
The third class are epimorphisms from a group onto its factor
group. Such homomorphisms can be constructed by
The fourth class is homomorphisms in a permutation group that are induced by an action on a set. Such homomorphisms are described in the context of group actions, see chapter Group Actions and in particular section ActionHomomorphism.
If map is a mapping from one group to another this attribute returns a group general mapping that which implements the same abstract mapping. (Some operations can be performed more effective in this representation, see also IsGroupGeneralMappingByAsGroupGeneralMappingByImages.)
gap> AsGroupGeneralMappingByImages(hom); [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,2) ]
37.2 Operations for Group Homomorphisms
Group homomorphisms are mappings, so all the operations and properties for mappings described in chapter Mappings are applicable to them. (However often much better methods, than for general mappings are available.)
Group homomorphisms will map groups to groups by just mapping the set of generators.
KernelOfMultiplicativeGeneralMapping can be used to compute the kernel
of a group homomorphism.
gap> hom:=GroupHomomorphismByImages(g,h,gens,[(1,2),(1,3)]);; gap> Kernel(hom); Group([ (1,2)(3,4), (1,3)(2,4) ])
Homomorphisms can map between groups in different representations and are also used to get isomorphic groups in a different representation.
gap> m1:=[[0,-1],[1,0]];;m2:=[[0,-1],[1,1]];; gap> sl2z:=Group(m1,m2);; # SL(2,Integers) as matrix group gap> F:=FreeGroup(2);; gap> psl2z:=F/[F.1^2,F.2^3]; #PSL(2,Z) as FP group <fp group on the generators [ f1, f2 ]> gap> phom:=GroupHomomorphismByImagesNC(sl2z,psl2z,[m1,m2], > GeneratorsOfGroup(psl2z)); # the non NC-version would be expensive [ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, 1 ] ] ] -> [ f1, f2 ] gap> Kernel(phom); # the diagonal matrices Group([ [ [ -1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ] ]) gap> p1:=(1,2)(3,4);;p2:=(2,4,5);;a5:=Group(p1,p2);; gap> ahom:=GroupHomomorphismByImages(psl2z,a5, > GeneratorsOfGroup(psl2z),[p1,p2]); # here homomorphism test is cheap. [ f1, f2 ] -> [ (1,2)(3,4), (2,4,5) ] gap> u:=PreImage(ahom,Group((1,2,3),(1,2)(4,5))); Group(<fp, no generators known>) gap> Index(psl2z,u); 10 gap> isofp:=IsomorphismFpGroup(u); [ f2*f1*f2^-1*f1^-1*f2^-1*f1^-1, f1*f2^-1*f1*f2*f1^-1, f2*f1*f2*f1*f2^-1*f1^-1*f2^-1, f2^-1*f1*f2*f1^-1*f2 ] -> [ _x1, _x2, _x3, _x4 ] gap> RelatorsOfFpGroup(Image(isofp)); [ _x2^2, _x3^2, _x4^3 ] gap> up:=PreImage(phom,u); <matrix group with 9 generators> gap> List(GeneratorsOfGroup(up),TraceMat); [ -2, -2, -2, -4, 0, -2, 0, 4, 1 ]
For an automorphism aut,
Inverse returns the inverse automorphism
aut -1. However if hom is a bijective homomorphism between
different groups, or if hom is injective and considered to be a bijection
to its image, the operation
InverseGeneralMapping should be used instead.
(See Inverse for a further discussion of this problem.)
gap> iso:=IsomorphismPcGroup(g); Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ f1, f2, f3, f4 ] gap> Inverse(iso); #I The mapping must be bijective and have source=range #I You might want to use `InverseGeneralMapping' fail gap> InverseGeneralMapping(iso); [ f1, f2, f3, f4 ] -> Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ])
37.3 Efficiency of Homomorphisms
GAP permits to create homomorphisms between arbitrary groups. This section considers the efficiency of the implementation and shows ways how to choose suitable representations. For permutation groups (see Permutation Groups) or Pc groups (see Pc Groups) this is normally nothing to worry about, unless the groups get extremely large. For other groups however certain calculations might be expensive and some precaution might be needed to avoid unnecessarily expensive calculations.
In short, it is always worth to tell a mapping that it is a homomorphism
(this can be done by
SetIsMapping) (or to create it direcetly with
The basic operations required are to compute image and preimage of elements and to test whether a mapping is a homomorphism. Their cost will differ depending on the type of the mapping.
Mappings given on generators (
Computing images requires to express an element of the source as word in the
generators. If it cannot be done effectivly (this is determined by
KnowsHowToDecompose, see KnowsHowToDecompose which returns
example for arbitrary permutation groups, for Pc groups or for finitely
presented groups with the images of the free generators) the span of the
generators has to be computed elementwise which can be very expensive and
Computing preimages adheres to the same rules with swapped r^oles of generators and their images.
The test whether a mapping is a homomorphism requires
the computation of a presentation for the source and evaluation of its
relators in the images of its generators. For larger groups this can be
GroupHomomorphismByImagesNC should be used if the mapping is
known to be a homomorphism.
Action homomorphisms (
The calculation of images is determined by the acting function used and
-- for large domains -- is often dominated by the search for the position of
an image in a list of the domain elements. This can be improved by sorting
this list if an efficient method for
< to compare elements of the domain
Once the images of a generating set are computed, computing preimages (which is
done via the
AsGroupGeneralMappingByImages) and computing the kernel
bahaves the same as for a
GroupHomomorphismByImages in a permutation
GAP will always assume that the acting function provided implements a proper group action and thus that the mapping is indeed a homomorphism.
Mappings given by functions (
Computing images is wholly determined by the function that performs the image calculation. If no function to compute preimages is given, computing preimages requires mapping every element of the source to find an element that maps to the requested image. This is time and memory consuming.
Testing whether a
GroupGeneralMappingByFunctions is a homomorphism would
require mapping all products of elements and thus should be avoided.
To compute the kernel of a homomorphism (unless the mapping is known to be injective) requires the capability to compute a presentation of the image and to evaluate the relators of this presentation in preimages of the presentations generators.
The calculation of the
ImagesSource) requires to map
a generating set of the source, testing surjectivity is a comparison for
equality with the range.
Testing injectivity is a test for triviality of the kernel.
The comparison of mappings is based on a lexicographic comparison of a sorted element list of the source. For groups this can be simplified:
returns the list of images of
list can be used to compare group homomorphisms. (The standard
comparison is to compare the image lists on the set of elements of the
source. If however x and y have the same images under a and b,
certainly all their products have. Therefore it is sufficient to test
this on the images of the smallest generators.)
37.4 Nice Monomorphisms
GAP contains very efficient algorithms for some special representations of groups (for example pc groups or permutation groups) while for other representations only slow generic methods are available. In this case it can be worthwhile to do all calculations rather in an isomorphic image of the group, which is in a ``better'' representation. The way to achieve this in GAP is via nice monomorphisms.
For this mechanism to work, of course there must be effective methods to
NiceMonomorphism on elements and to take preimages under it.
As by definition no good algorithms exist for the source group,
normally this can only be achieved by using an
ActionHomomorphism or a
GroupHomomorphismByFunction (see also
section Efficiency of Homomorphisms).
If this property is
true, high-valued methods that translate all
calculations in obj in the image under the
available for obj.
is a homomorphism that is defined (at least) on the whole of obj and whose restriction to obj is injective. The concrete morphism (and also the image group) will depend on the representation of obj.
NiceObject of obj is the image of obj under its
A typical example are finite matrix groups, which use a faithful action on vectors to translate all calculations in a permutation group.
gap> gl:=GL(3,2); SL(3,2) gap> IsHandledByNiceMonomorphism(gl); true gap> NiceObject(gl); Group([ (5,7)(6,8), (2,3,5)(4,7,6) ]) gap> Image(NiceMonomorphism(gl),Z(2)*[[1,0,0],[0,1,1],[1,0,1]]); (2,6)(3,4,7,8)
NiceMonomorphism nhom is canonical if the image set will only
depend on the set of group elements but not on the generating set and
< comparison of group elements translates through the nice
implies that equal objects will always have equal
In some situations however this condition would be expensive to
achieve, therefore it is not guaranteed for every nice monomorphism.
37.5 Group Automorphisms
Group automorphisms are bijective homomorphism from a group onto itself. An important subclass are automorphisms which are induced by conjugation of the group itself or a supergroup.
Let G be a group, and g an element in the same family as the elements
ConjugatorIsomorphism returns the isomorphism from G to
defined by h ® h g for all h Î G .
If g normalizes G then
ConjugatorIsomorphism does the same as
ConjugatorAutomorphismNC (see ConjugatorAutomorphism).
Let G be a group, and g an element in the same family as the elements
of G such that g normalizes G.
ConjugatorAutomorphism returns the automorphism of G
defined by h ® h g for all h Î G .
If conjugation by g does not leave G invariant,
in this case,
the isomorphism from G to
induced by conjugation with g
can be constructed
ConjugatorIsomorphism (see ConjugatorIsomorphism).
ConjugatorAutomorphismNC does the same as
except that the check is omitted whether g normalizes G.
Let G be a group, and g Î G .
InnerAutomorphism returns the automorphism of G
defined by h ® h g for all h Î G .
If g is not an element of G,
in this case,
the isomorphism from G to
induced by conjugation with g
can be constructed
ConjugatorIsomorphism (see ConjugatorIsomorphism)
ConjugatorAutomorphism (see ConjugatorAutomorphism).
InnerAutomorphismNC does the same as
except that the check is omitted whether g Î G .
Let hom be a group general mapping (see IsGroupGeneralMapping)
with source G, say.
true if hom is induced by
conjugation of G by an element g that lies in G or in a group into
which G is naturally embedded in the sense described below,
Natural embeddings are dealt with in the case that G is
a permutation group (see Chapter Permutation Groups),
a matrix group (see Chapter Matrix Groups),
a finitely presented group (see Chapter Finitely Presented Groups), or
a group given w.r.t. a polycyclic presentation (see Chapter Pc Groups).
In all other cases,
IsConjugatorIsomorphism may return
if hom is induced by conjugation but is not an inner automorphism.
true for hom then
an element g that induces hom can be accessed as value of
true if hom is an automorphism
(see IsEndoGeneralMapping) that is regarded as a conjugator isomorphism
true if hom is a conjugator
automorphism such that an element g inducing hom can be chosen in
For a conjugator isomorphism hom (see ConjugatorIsomorphism),
ConjugatorOfConjugatorIsomorphism returns an element g such that
mapping under hom is induced by conjugation with g.
To avoid problems with
it is guaranteed that the conjugator is taken from the source of hom
gap> hgens:=[(1,2,3),(1,2,4)];;h:=Group(hgens);; gap> hom:=GroupHomomorphismByImages(h,h,hgens,[(1,2,3),(2,3,4)]);; gap> IsInnerAutomorphism(hom); true gap> ConjugatorOfConjugatorIsomorphism(hom); (1,2,3) gap> hom:=GroupHomomorphismByImages(h,h,hgens,[(1,3,2),(1,4,2)]); [ (1,2,3), (1,2,4) ] -> [ (1,3,2), (1,4,2) ] gap> IsInnerAutomorphism(hom); false gap> IsConjugatorAutomorphism(hom); true gap> ConjugatorOfConjugatorIsomorphism(hom); (1,2)
37.6 Groups of Automorphisms
Group automorphissm can be multiplied and inverted and thus it is possible to form groups of automorphisms.
indicates whether G consists of automorphisms of another group H.
The group H can be obtained from G via the attribute
If G consists of automorphisms of H, this attribute returns H.
returns the full automorphism group of the object obj. The
automorphisms act on the domain by the caret operator
The automorphism group often stores a ``NiceMonomorphism'' (see
NiceMonomorphism) to a permutation group, obtained by the action on a
subset of obj.
indicates whether G is the full automorphism group of another group
H, this group is given as
AutomorphismDomain of G.
gap> g:=Group((1,2,3,4),(1,3)); Group( [ (1,2,3,4), (1,3) ] ) gap> au:=AutomorphismGroup(g); <group of size 8 with 3 generators> gap> GeneratorsOfGroup(au); [^(1,2,3,4),^(1,3),[(2,4),(1,2,3,4)]->[(1,2)(3,4),(1,4,3,2)]] gap> NiceObject(au); Group([(1,4)(2,6),(2,6)(3,5),(1,2)(3,5)(4,6)])
For an automorphism group autgroup of a group this attribute stores the subgroup of inner automorphisms (automorphisms induced by conjugation) of the original group.
gap> InnerAutomorphismsAutomorphismGroup(au); < group with 2 generators>
Let aut be an automorphism of a group G and epi > :G - H an homomorphism such that kerepi is fixed under aut. Let U be the image of epi. This command returns the automorphism of U induced by aut via epi, that is the automorphism of U which maps gepi to (gaut )epi .
gap> g:=Group((1,2,3,4),(1,2)); Group( [ (1,2,3,4), (1,2) ] ) gap> n:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]); Group( [ (1,2)(3,4), (1,3)(2,4) ] ) gap> epi:=NaturalHomomorphismByNormalSubgroup(g,n); [ (3,4), (2,4,3), (1,2)(3,4), (1,3)(2,4) ] -> [ f1, f2, <identity> of ..., <identity> of ... ] gap> aut:=InnerAutomorphism(g,(1,2,3)); ^(1,2,3) gap> InducedAutomorphism(epi,aut); ^f2
37.7 Calculating with Group Automorphisms
Usually the best way to calculate in a group of automorphisms is to go
translate all calculations to an isomorphic group in a representation, for
which better algorithms are available, say a permutation group. This
translation can be done automatically using a
Once a group knows to be a group of automorphisms (this can be achieved
by testing or setting the property
GAP will try itself to find such a nice monomorphism once calculations in
the automorphism group are done.
computes a nice monomorphism for autgroup acting on group and stores
NiceMonomorphism in autgrp.
If the centre of
AutomorphismDomain of autgrp is trivial, the
operation will first try to represent all automorphisms by conjugation
(in group or a natural parent of group).
If this fails the operation tries to find a small subset of group on which the action will be faithful.
The operation sets the attribute
NiceMonomorphism and does not return
If a good domain for a faithful permutation action is know already, a
homomorphism for the action on it can be created using
NiceMonomorphismAutomGroup. It might be stored by
This function creates a monomorphism for an automorphism group
autgrp of a group by permuting the group elements in the list elms.
This list must be chosen to yield a faithful representation. elmsgens
is a list of generators which are a subset of elms. (They can differ
from the groups original generators.) It does not yet assign it as
Another nice way of representing automorphisms as permutations has been described in Sims97. It it not yet available in GAP, a description however can be found in section Stabilizer Chains for Automorphisms Acting on Enumerators of ``Extending GAP''.
37.8 Searching for Homomorphisms
computes an isomorphism between the groups G and H
if they are isomorphic and returns
gap> g:=Group((1,2,3,4),(1,3));; gap> h:=Group((1,4,6,7)(2,3,5,8), (1,5)(2,6)(3,4)(7,8));; gap> IsomorphismGroups(g,h); [ (1,2,3,4), (1,3) ] -> [ (1,4,6,7)(2,3,5,8), (1,2)(3,7)(4,8)(5,6) ] gap> IsomorphismGroups(g,Group((1,2,3,4),(1,2))); fail
computes all epimorphisms from F onto G up to automorphisms of G. This classifies all factor groups of F which are isomorphic to G.
gap> g:=Group((1,2,3,4),(1,2)); Group( [ (1,2,3,4), (1,2) ] ) gap> h:=Group((1,2,3),(1,2)); Group( [ (1,2,3), (1,2) ] ) gap> quo:=GQuotients(g,h); [[(1,3,2,4),(1,3,4)]->[(2,3),(1,2,3)]]
computes all monomorphisms from H onto G up to G-conjugacy of the image groups. This classifies all G-classes of subgroups of G which are isomorphic to H.
gap> g:=Group((1,2,3,4),(1,2)); Group( [ (1,2,3,4), (1,2) ] ) gap> h:=Group((3,4),(1,2));; gap> emb:=IsomorphicSubgroups(g,h); [ [ (3,4), (1,2) ] -> [ (1,4)(2,3), (1,3)(2,4) ], [ (3,4), (1,2) ] -> [ (1,2)(3,4), (3,4) ] ]
This function loops over element tuples taken from classes and checks these for properties like generating a given group of fulfilling relations. This can be used to find small generating sets or all types of Morphisms. The element tuples are classified only up to up to inner automorphisms as all images can be obtained easily from them by conjugation but usually running through all of them would take too long.
range is a groups containing the elements.
The classes are given in a list classes which is a list of records
like the ones returned from
params is a record containing optional components:
gens. They are given as a list [word,order] where word is a word in the free generators
action is a number whose bit-representation indicates the requirements
which are enforced on the element tuples found:parnoindent
8 find all (otherwise stops after the first find)parnoindent
If the search is for homomorphisms, the function returns homomorphisms
obtained by mapping the given generators
gens instead of element tuples.
The ``Morpheus'' algorithm used to find homomorphisms is described in section V.5 of Hulpke96.
37.9 Representations for Group Homomorphisms
The different represntations of group homomorphisms are used to indicate from what type of group to what type of group they map and thus determine which methods are used to compute images and preimages.
The information in this section is mainly relevant for implementing new methods and not for using homomorphisms.
Representation for mappings from one group to another that are defined by extending a mapping of group generators homomorphically.
Representation for mappings that delegate work on a
Representation for mappings that delegate work for preimages to a GroupHomomorphismByImages.
is the representation for mappings that map from a perm group
is the representation for mappings that map to a perm group
is the representations for mappings that map a pcgs to images and thus may use exponents to decompose generators.
is the representation for mappings from a pc group
is the representation for mappings to a pc group
is the representation of mappings from an fp group.
is the representation of mappings from an fp group that give images of the standard generators.
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GAP 4 manual