37.8 Searching for Homomorphisms

computes an isomorphism between the groups G and H if they are isomorphic and returns fail otherwise.

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 MorMaxFusClasses. params is a record containing optional components:

gens

generators that are to be mapped (for testing morphisms). The length of this list determines the length of element tuples considered.

from

a preimage group (that contains gens)

to

image group (which might be smaller than range)

free

free generators, a list of the same length than the generators gens.

rels

some relations that hold among the generators gens. They are given as a list [word,order] where word is a word in the free generators free.

dom

a set of elements on which automorphisms act faithfully (used to do element tests in partial automorphism groups).

aut

Subgroup of already known automorphisms.

action is a number whose bit-representation indicates the requirements which are enforced on the element tuples found:parnoindent 1  homomorphismparnoindent 2  injectiveparnoindent 4  surjectiveparnoindent 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.

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