%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A matgrp.tex GAP documentation Martin Schoenert %% %A @(#)$Id: matgrp.tex,v 3.5 1993/03/11 17:56:24 fceller Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes matrix group and their implementation. %% %H $Log: matgrp.tex,v $ %H Revision 3.5 1993/03/11 17:56:24 fceller %H added matrix group record %H %H Revision 3.4 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.3 1993/02/12 16:44:35 felsch %H examples adjusted to line length 72 %H %H Revision 3.2 1993/02/05 10:51:35 felsch %H example fixed %H %H Revision 3.1 1992/04/04 15:56:52 martin %H initial revision under RCS %H %% \Chapter{Matrix Groups} A *matrix group* is a group of invertable square matrices (see chapter "Matrices"). In {\GAP} you can define matrix groups of matrices over each of the fields that {\GAP} supports, i.e., the rationals, cyclotomic extensions of the rationals, and finite fields (see chapters "Rationals", "Cyclotomics", and "Finite Fields"). You define a matrix group in {\GAP} by calling 'Group' (see "Group") passing the generating matrices as arguments. | gap> m1 := [ [ Z(3)^0, Z(3)^0, Z(3) ], > [ Z(3), 0*Z(3), Z(3) ], > [ 0*Z(3), Z(3), 0*Z(3) ] ];; gap> m2 := [ [ Z(3), Z(3), Z(3)^0 ], > [ Z(3), 0*Z(3), Z(3) ], > [ Z(3)^0, 0*Z(3), Z(3) ] ];; gap> m := Group( m1, m2 ); Group( [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ] ], [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), Z(3) ] ] )| However, currently {\GAP} can only compute with finite matrix groups. Also computations with large matrix groups are not done very efficiently. We hope to improve this situation in the future, but currently you should be careful not to try too large matrix groups. Because matrix groups are just a special case of domains all the set theoretic functions such as 'Size' and 'Intersection' are applicable to matrix groups (see chapter "Domains" and "Set Functions for Matrix Groups"). Also matrix groups are of course groups, so all the group functions such as 'Centralizer' and 'DerivedSeries' are applicable to matrix groups (see chapter "Groups" and "Group Functions for Matrix Groups"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Set Functions for Matrix Groups} \index{in!for matrix groups} \index{Size!for matrix groups} \index{Intersection!for matrix groups} \index{Random!for matrix groups} As already mentioned in the introduction of this chapter matrix groups are domains. All set theoretic functions such as 'Size' and 'Intersections' are thus applicable to matrix groups. This section describes how these functions are implemented for matrix groups. Functions not mentioned here either inherit the default group methods described in "Set Functions for Groups" or the default method mentioned in the respective sections. To compute with a matrix group , {\GAP} computes the operation of the matrix group on the underlying vector space (more precisely the union of the orbits of the parent of on the standard basis vectors). Then it works with the thus defined permutation group

, which is of course isomorphic to , and finally translates the results back into the matrix group. \vspace{5mm} ' in ' To test if an object lies in a matrix group , {\GAP} first tests whether is a invertable square matrix of the same dimensions as the matrices of . If it is, {\GAP} tests whether permutes the vectors in the union of the orbits of on the standard basis vectors. If it does, {\GAP} takes this permutation and tests whether it lies in

. \vspace{5mm} 'Size( )' To compute the size of the matrix group , {\GAP} computes the size of the isomorphic permutation group

. \vspace{5mm} 'Intersection( , )' To compute the intersection of two subgroups and with a common parent matrix group , {\GAP} intersects the images of the corresponding permutation subgroups and of

. Then it translates the generators of the intersection of the permutation subgroups back to matrices. The intersection of and is the subgroup of generated by those matrices. If and do not have a common parent group, or if only one of them is a matrix group and the other is a set of matrices, the default method is used (see "Intersection"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Group Functions for Matrix Groups} \index{Centralizer!for matrix groups} \index{Normalizer!for matrix groups} \index{SylowSubgroup!for matrix groups} \index{ConjugacyClasses!for matrix groups} \index{PermGroup!for matrix groups} \index{Stabilizer!for matrix groups} \index{RepresentativeOperation!for matrix groups} As already mentioned in the introduction of this chapter matrix groups are after all group. All group functions such as 'Centralizer' and 'DerivedSeries' are thus applicable to matrix groups. This section describes how these functions are implemented for matrix groups. Functions not mentioned here either inherit the default group methods described in the respective sections. To compute with a matrix group , {\GAP} computes the operation of the matrix group on the underlying vector space (more precisely the union of the orbits of the parent of on the standard basis vectors). Then it works with the thus defined permutation group

, which is of course isomorphic to , and finally translates the results back into the matrix group. \vspace{5mm} 'Centralizer( , )' \\ 'Normalizer( , )' \\ 'SylowSubgroup( ,

)' \\ 'ConjugacyClasses( )' This functions all work by solving the problem in the permutation group

and translating the result back. \vspace{5mm} 'PermGroup( )' This computes a permutation representation of by letting operate on the union of the orbits of (*not* its parent) on the standard basis vectors. \vspace{5mm} 'Stabilizer( , )' The stabilizer of a vector that lies in the union of the orbits of the parent of on the standard basis vectors is computed by finding the stabilizer of the corresponding point in the permutation group

and translating this back. Other stabilizers are computed with the default method (see "Stabilizer"). \vspace{5mm} 'RepresentativeOperation( , , )' If and are vectors that both lie in the union of the orbits of the parent group of on the standard basis vectors, 'RepresentativeOperation' finds a permutation in

that takes the point corresponding to to the point corresponding to . If no such permutation exists, it returns 'false'. Otherwise it translates the permutation back to a matrix. \vspace{5mm} 'RepresentativeOperation( , , )' If and are matrices in , 'RepresentativeOperation' finds a permutation in

that conjugates the permutation corresponding to to the permutation corresponding to . If no such permutation exists, it returns 'false'. Otherwise it translates the permutation back to a matrix. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Matrix Group Records} A group is represented by a record that contains information about the group. A matrix group record contains the following components in addition to those described in section "Group Records". 'isMatGroup': \\ always 'true'. If a permutation representation for a matrix group is known it is stored in the following components. 'permGroupP': \\ contains the permutation group representation of . 'permDomain': \\ contains the union of the orbits of the parent of on the standard basis vectors. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%