Introduction ============ GAP is a system for computational discrete algebra, which we have developed with particular emphasis on computational group theory, but which has already proved useful also in other areas. The name GAP is an acronym for *Groups, Algorithms, and Programming*. This (long) document announces the availability of GAP version 3 release 1, GAP 3.1 for short. It is an *advertisement* for GAP, but not a *commercial*, since we give GAP away for free. This document begins with the section "Announcement", which contains the announcement proper. The next section "Analyzing Rubik's Cube with GAP" contains an extensive example. This example is followed by a general discussion of GAP's capabilities in the section "An Overview of GAP". The next sections "How to get GAP" and "How to install GAP" describe how you can get GAP running on your computer. Then we tell you about our plans for the future in the section "The Future of GAP". The final section "The GAP Forum" introduces the GAP forum, where interested users can discuss GAP related topics by e-mail messages. Announcement ============ Il est trop tard, maintenant, il sera toujours trop tard. Heureusement! (A. Camus, La chute) ######## Lehrstuhl D fuer Mathematik ### #### RWTH Aachen ## ## ## # ####### ######### ## # ## ## # ## ## # # ## # ## #### ## ## # # ## ##### ### ## ## ## ## ######### # ######### ####### # # ## Version 3 # ### Release 1 # ## # 7 Apr 92 # ## # ## # Johannes Meier, Martin Schoenert ## # Alice Niemeyer, Werner Nickel ## # Alex Wegner, Thomas Bischops ### ## Juergen Mnich, Frank Celler ###### Thomas Breuer, Goetz Pfeiffer Udo Polis Lehrstuhl D f"ur Mathematik, RWTH Aachen, announces the availability of GAP version 3 release 1, or GAP 3.1 for short. This is the first publicly available release of GAP since version 2.4, which was distributed since January 1989. Analyzing Rubik's Cube with GAP =============================== Ideal Toy Company stated on the package of the original Rubik cube that there were more than three billion possible states the cube could attain. It's analogous to Mac Donald's proudly announcing that they've sold more than 120 hamburgers. (J. A. Paulos, Innumeracy) To show you what GAP can do a short example is probably best. If you are not interested in this example skip to the section "An Overview of GAP". For the example we consider the group of transformations of Rubik's magic cube. If we number the faces of this cube as follows +--------------+ | 1 2 3 | | 4 top 5 | | 6 7 8 | +--------------+--------------+--------------+--------------+ | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | | 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 | | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | +--------------+--------------+--------------+--------------+ | 41 42 43 | | 44 bottom 45 | | 46 47 48 | +--------------+ then the group is generated by the following generators, corresponding to the six faces of the cube (the two semicolons tell GAP not to print the result, which is identical to the input here). gap> cube := Group( > ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19), > ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35), > (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11), > (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24), > (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27), > (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) > );; First we want to know the size of this group. gap> Size( cube ); 43252003274489856000 Since this is a little bit unhandy, let us factorize this number. gap> Collected( Factors( last ) ); [ [ 2, 27 ], [ 3, 14 ], [ 5, 3 ], [ 7, 2 ], [ 11, 1 ] ] (The result tells us that the size is 2^27 3^14 5^3 7^2 11.) Next let us investigate the operation of the group on the 48 points. gap> orbits := Orbits( cube, [1..48] ); [ [ 1, 3, 17, 14, 8, 38, 9, 41, 19, 48, 22, 6, 30, 33, 43, 11, 46, 40, 24, 27, 25, 35, 16, 32 ], [ 2, 5, 12, 7, 36, 10, 47, 4, 28, 45, 34, 13, 29, 44, 20, 42, 26, 21, 37, 15, 31, 18, 23, 39 ] ] The first orbit contains the points at the corners, the second those at the edges; clearly the group cannot move a point at a corner onto a point at an edge. So to investigate the cube group we first investigate the operation on the corner points. Note that the constructed group that describes this operation will operate on the set [1..24], not on the original set [1,3,17,14,8,38,9,41,19,48,22,6,30,33,43,11,46,40,24,27,25,35,16,32]. gap> cube1 := Operation( cube, orbits[1] ); Group( ( 1, 2, 5,12)( 3, 7,14,21)( 9,16,22,20), ( 1, 3, 8,18)( 4, 7,16,23)(11,17,22,12), ( 3, 9,19,11)( 5,13, 8,16)(12,21,15,23), ( 2, 6,15, 9)( 5,14,10,19)(13,21,20,24), ( 1, 4,10,20)( 2, 7,17,24)( 6,14,22,18), ( 4,11,13, 6)( 8,15,10,17)(18,23,19,24) ) gap> Size( cube1 ); 88179840 Now this group obviously operates transitively, but let us test whether it is also primitive. gap> corners := Blocks( cube1, [1..24] ); [ [ 1, 7, 22 ], [ 2, 14, 20 ], [ 3, 12, 16 ], [ 4, 17, 18 ], [ 5, 9, 21 ], [ 6, 10, 24 ], [ 8, 11, 23 ], [ 13, 15, 19 ] ] Those eight blocks correspond to the eight corners of the cube; on the one hand the group permutes those and on the other hand it permutes the three points at each corner cyclically. So the obvious thing to do is to investigate the operation of the group on the eight corners. gap> cube1b := Operation( cube1, corners, OnSets ); Group( (1,2,5,3), (1,3,7,4), (3,5,8,7), (2,6,8,5), (1,4,6,2), (4,7,8,6) ) gap> Size( cube1b ); 40320 Now a permutation group of degree 8 that has order 40320 must be the full symmetric group S(8) on eight points. The next thing then is to investigate the kernel of this operation on blocks, i.e., the subgroup of 'cube1' of those elements that fix the blocks setwise. gap> blockhom1 := OperationHomomorphism( cube1, cube1b );; gap> Factors( Size( Kernel( blockhom1 ) ) ); [ 3, 3, 3, 3, 3, 3, 3 ] gap> IsElementaryAbelian( Kernel( blockhom1 ) ); true We can show that the product of this elementary abelian group 3^7 with the S(8) is semidirect by finding a complement, i.e., a subgroup that has trivial intersection with the kernel and that generates 'cube1' together with the kernel. gap> cmpl1 := Stabilizer( cube1, [1,2,3,4,5,6,8,13], OnSets );; gap> Size( cmpl1 ); 40320 gap> Size( Intersection( cmpl1, Kernel( blockhom1 ) ) ); 1 gap> Closure( cmpl1, Kernel( blockhom1 ) ) = cube1; true There is even a more elegant way to show that 'cmpl1' is a complement. gap> IsIsomorphism( OperationHomomorphism( cmpl1, cube1b ) ); true Of course, theoretically it is clear that 'cmpl1' must indeed be a complement. In fact we know that 'cube1' is a subgroup of index 3 in the wreath product of a cyclic 3 with S(8). This missing index 3 tells us that we do not have total freedom in turning the corners. The following tests show that whenever we turn one corner clockwise we must turn another corner counterclockwise. gap> (1,7,22) in cube1; false gap> (1,7,22)(2,20,14) in cube1; true More or less the same things happen when we consider the operation of the cube group on the edges. gap> cube2 := Operation( cube, orbits[2] );; gap> Size( cube2 ); 980995276800 gap> edges := Blocks( cube2, [1..24] ); [ [ 1, 11 ], [ 2, 17 ], [ 3, 19 ], [ 4, 22 ], [ 5, 13 ], [ 6, 8 ], [ 7, 24 ], [ 9, 18 ], [ 10, 21 ], [ 12, 15 ], [ 14, 20 ], [ 16, 23 ] ] gap> cube2b := Operation( cube2, edges, OnSets );; gap> Size( cube2b ); 479001600 gap> blockhom2 := OperationHomomorphism( cube2, cube2b );; gap> Factors( Size( Kernel( blockhom2 ) ) ); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] gap> IsElementaryAbelian( Kernel( blockhom2 ) ); true gap> cmpl2 := Stabilizer(cube2,[1,2,3,4,5,6,7,9,10,12,14,16],OnSets);; gap> IsIsomorphism( OperationHomomorphism( cmpl2, cube2b ) ); true This time we get a semidirect product of a 2^11 with an S(12), namely a subgroup of index 2 of the wreath product of a cyclic 2 with S(12). Here the missing index 2 tells us again that we do not have total freedom in turning the edges. The following tests show that whenever we flip one edge we must also flip another edge. gap> (1,11) in cube2; false gap> (1,11)(2,17) in cube2; true Since 'cube1' and 'cube2' are the groups describing the actions on the two orbits of 'cube', it is clear that 'cube' is a subdirect product of those groups, i.e., a subgroup of the direct product. Comparing the sizes of 'cube1', 'cube2', and 'cube' we see that 'cube' must be a subgroup of index 2 in the direct product of those two groups. gap> Size( cube ); 43252003274489856000 gap> Size( cube1 ) * Size( cube2 ); 86504006548979712000 This final missing index 2 tells us that we cannot operate on corners and edges totally independently. The following tests show that whenever we exchange a pair of corners we must also exchange a pair of edges (and vice versa). gap> (17,19)(11,8)(6,25) in cube; false gap> (7,28)(18,21) in cube; false gap> (17,19)(11,8)(6,25)(7,28)(18,21) in cube; true Finally let us compute the centre of the cube group, i.e., the subgroup of those operations that can be performed either before or after any other operation with the same result. gap> Centre( cube ); Subgroup( cube, [ ( 2,34)( 4,10)( 5,26)( 7,18)(12,37)(13,20) (15,44)(21,28)(23,42)(29,36)(31,45)(39,47) ] ) We see that the centre contains one nontrivial element, namely the operation that flips all 12 edges simultaneously. This concludes our example. Of course, GAP can do much more, and the next section gives an overview of its capabilities, but demonstrating them all would take too much room. An Overview of GAP ================== Though this be madness, yet there is method in't. (W. Shakespeare, Hamlet) GAP consists of several parts: the kernel, the library of functions, the library of groups and related data, and the documentation. The *kernel* implements an automatic memory management, a PASCAL-like programming language, also called GAP, with special datatypes for computations in group theory, and an interactive programming environment to run programs written in the GAP programming language. The automatic *memory management* allows programmers to concentrate on implementing the algorithm without needing to care about allocation and deallocation of memory. It includes a garbage collection that automatically throws away objects that are no longer accessible. The GAP programming language supports a number of datatypes for elements of fields. *Integers* can be arbitrarily large, and are implemented in such a way that operations with small integers are reasonably fast. Building on this large-integer arithmetic GAP supports *rationals* and elements from *cyclotomic fields*. Also GAP allows one to work with elements from *finite fields* of size (at present) at most 2^16. The special datatypes of group elements are *permutations* (currently operating on at most 2^16 points, but we are going to remove this restriction soon), *matrices* over the rationals, cyclotomic fields, and finite fields, *words in abstract generators*, and *words in solvable groups*. GAP also contains a very flexible *list* datatype. A list is simply a collection of objects that allows you to access the components using an integer position. Lists grow automatically when you add new elements to them. Lists are used to represent sets, vectors, and matrices. A *set* is represented by a sorted list without duplicates. A list whose elements all lie in a common field is a *vector*. A list of vectors of the same length over a common field is a *matrix*. Since sets, vectors, and matrices are lists, all list operations and functions are applicable. You can, for example, find a certain element in a vector with the general function 'Position'. There are also *ranges*, i.e., lists of consecutive integers, and *boolean lists*, i.e., lists containing only 'true' and 'false'. Vectors, ranges, and boolean lists have special internal representations to ensure efficient operations and memory usage. For example, a boolean list requires only one bit per element. *Records* in GAP are similar to lists, except that accessing the components of a record is done using a name instead of an index. Records are used to collect objects of different types, while lists usually only contain elements of one type. Records are for example used to represent groups and other domains; there is *no* group datatype in the GAP language . Because of this all information that GAP knows about a group is also accessible to you by simply investigating the record. The control structures of GAP are PASCAL-like. GAP has *if* statements, *while*, *repeat*, and *for* loops. The for loop is a little bit uncommon in that it always loops over the elements of a list. The usual semantics can be obtained by looping over the elements of a range. Using those building blocks you can write *functions*. Functions can be recursive, and are first class objects in the sense that you can collect functions in lists, pass them as arguments to other functions and also return them. It is important to note that GAP has dynamic typing instead of static typing. That means that the datatype is a property of the object, not of the variable. This allows you to write general functions. For example the generic function that computes an orbit can be used to compute the orbit of an integer under a permutation group, the orbit of a vector under a matrix group, the conjugacy class of a group element, and many more. The kernel also implements an *interactive environment* that allows you to use GAP. This environment supports debugging; in case of an error a break loop is entered in which you can investigate the problem, and maybe correct it and continue. You also have online access to the manual, though sections that contain larger formulas do not look nice on the screen. The *library of functions*, simply called library in the following, contains implementations of various group theoretical algorithms written in the GAP language. Because all the group theoretical functions are in this library it is easy for you to look at them to find out how they work, and change them if they do almost, but not quite, what you want. The whole library is centered around the concept of domains and categories. A *domain* is a structured set, e.g., a group is a domain as is the ring of Gaussian integers. Each domain in GAP belongs to one or more *categories*, which are simply sets of domains, e.g., the set of all groups forms a category. The categories in which a domain lies determine the functions that are applicable to this domain and its elements. To each domain belongs a set of functions, in a so called operations record, that are called by dispatchers like 'Size'. For example, for a permutation group , '.operations.Size' is a function implementing the Schreier Sims algorithm. Thus if you have any domain , simply calling 'Size( )' will return the size of the domain , computed by an appropriate function. Domains *inherit* such functions from their category, unless they redefine them. For example, for a permutation group , the derived subgroup will be computed by the generic group function, which computes the normal closure of the subgroup generated by the commutators of the generators. Of course the most important category is the category of *groups*. There are about 100 functions applicable to groups. These include general functions such as 'Centralizer' and 'SylowSubgroup', functions that compute series of subgroups such as 'LowerCentralSeries', a function that computes the whole lattice of subgroups, functions that test predicates such as 'IsSimple', functions that are related to the operations of groups such as 'Stabilizer', and many more. Most of these functions are applicable to all groups, e.g., permutation groups, finite polycyclic groups, factor groups, direct products of arbitrary groups, and even new types of groups that you create by simply specifying how the elements are multiplied and inverted (actually it is not quite so simple, but you can do it). Where the general functions that are applicable to all groups are not efficient enough, we have tried to overlay them by more efficient functions for special types of groups. The prime example is the category of *permutation groups*, which overlays 'Size', 'Elements', 'Centralizer', 'Normalizer', 'SylowSubgroup', and a few more functions by functions that employ stabilizer chains and backtracking algorithms. Also many of the functions that deal with operations of groups are overlayed for permutation groups for the operation of a permutation group on integers or lists of integers. For *finitely presented groups* only some functions are as yet implemented. They include finding the index of a subgroup via a Todd-Coxeter coset enumeration, computing the abelian invariants of the commutator factor group, intersecting two subgroups, finding the normalizer of a subgroup, and finding all subgroups of small index. Of course it is possible to go to a permutation group operating on the cosets of a subgroup and then to work with this permutation group. Clearly there is still much to be done in this area, most notably computing presentations for subgroups and simplifying presentations. We hope to be able to add such functions in the near future. For *finite polycyclic groups* a special kind of presentation corresponding to a composition series is used. Such a presentation implies a canonical form for the elements and thus allows efficient operations with the elements of such a group. This presentation is used to make functions such as 'Centralizer', 'Normalizer', 'Intersection', and 'ConjugacyClasses' very efficient. GAP's capabilities for finite polycyclic groups exceed those of the computer system SOGOS (which was developed at Lehrstuhl D f"ur Mathematik for the last decade). There is also support for *mappings* and *homomorphisms*. Since they play such a ubiquitous role in mathematics, it is only natural that they should also play an important role in a system like GAP. Mappings and homomorphisms are objects in their own right in GAP. You can apply a mapping to an element of its source, multiply mappings (provided that the range of the first is a subset of the source of the second), invert mappings (even if what you get is a multi-valued mapping), and perform a few more operations. Important examples are the 'NaturalHomomorphism' onto a factor group, 'OperationsHomomorphism' mapping a group that operates on a set of elements into the symmetric group on [1..], 'Embeddings' into products of groups, 'Projections' from products of groups onto the components, and the general 'GroupHomomorphismByImages' for which you only specify the images of a set of generators. The library contains a package for handling character tables of finite groups. This includes almost all possibilities of the computer system CAS (which was developed at Lehrstuhl D f"ur Mathematik in the last decade), and many new functions. You can compute character tables of groups, or construct character tables using other tables, or do some calculations within known character tables. You can, for example, compute a list of candidates for permutation characters. Of course there are many character tables (at the moment more than 650 ordinary tables) in the data library, including all those in the ATLAS of finite groups. For large integers we now also have a package for *elementary number theory*. There are functions in this package to test primality, factor integers of reasonable size, compute the size phi() of the prime residue group modulo an integer , compute roots modulo an integer , etc. Also based on this there is a package to do calculations in the ring of Gaussian integers. The library also includes a package for *combinatorics*. This contains functions to find all selections of various flavours of the elements of a set, e.g., 'Combinations' and 'Tuples', or the number of such selections, e.g., 'Binomial'. Other functions are related to partitions of sets or integers, e.g., 'PartitionsSet' and 'RestrictedPartitions', or the number of such, e.g., 'NrPartitions' and 'Bell'. It also contains some miscellaneous functions such as 'Fibonacci' and 'Bernoulli'. The *data library* at present contains the primitive permutation groups of degree up to 50 from C. Sims, the 2-groups of size dividing 256 from E. O'Brien and M. F. Newman, the solvable groups of size up to 100 from M. Hall, J. K. Senior, R. Laue, and J. Neub"user, and a library of character tables including all of the ATLAS. We plan to extend the data library with more data in the future. How to get GAP ============== Ceterum censeo: Nobody has ever paid a licence fee for using a proof that shows Sylow's subgroups to exist. Nobody should ever pay a licence fee for using a program that computes Sylow's subgroups. (J. Neub"user) GAP is distributed *free of charge*. You can obtain it via 'ftp' or electronic mail and give it away to your colleagues. GAP is *not* in the public domain, however. In particular you are not allowed to incorporate GAP or parts thereof into a commercial product. If you get GAP, we would appreciate it if you could notify us, e.g., by sending a short e-mail message to 'gap@samson.math.rwth-aachen.de', containing your full name and address, so that we have a rough idea of the number of users. We also hope that this number will be large enough to convince various agencies that GAP is a project worthy of (financial) support. If you publish some result that was partly obtained using GAP, we would appreciate it if you would cite GAP, just as you would cite another paper that you used. Again we would appreciate if you could inform us about such a paper. We distribute the *full source* for everything, the C code for the kernel, the GAP code for the library, and the LaTeX code for the manual, which has at present about 700 pages. So it should be no problem to get GAP, even if you have a rather uncommon system. Of course, ports to non UNIX systems may require some work. We already have a port to the Atari ST. A port to IBM PC compatibles with an Intel 80386 or 80486 is under way. We also hope to provide a port to the Apple Macintosh in the near future. Note that about 4 MByte of main memory and a harddisk are required to run GAP. GAP 3.1 can be obtained by anonymous *ftp* from the following servers. 'samson.math.rwth-aachen.de': Lehrstuhl D fur Mathematik, RWTH Aachen, Germany (137.226.152.6). 'dimacs.rutgers.edu': DIMACS, Rutgers, New Brunswick, New Jersey (128.6.75.16). 'math.ucla.edu': Math. Dept., Univ. of California at Los Angeles (128.97.4.254). 'pell.anu.edu.au': Math. Research Section, Australian National Univ. (150.203.15.5). 'ftp' to the server *closest* to you, login as user 'ftp' and give your full e-mail address as password. GAP is in the directory 'pub/gap'. Remember when you transmit the files to set the file transfer type to *binary image*, otherwise you will only receive unusable garbage. Those servers will always have the latest version of GAP available. GAP can also be obtained via *electronic mail*. To get one of the files mentioned below send a message to 'listserv@samson.math.rwth-aachen.de' containing a line 'get GAP ', e.g., 'get GAP src3r1.tar.Z'. 'listserv' will reply by sending you the file as e-mail message. Because most files are large binary files they will be uuencoded and split into several parts, each at most 64 kBytes large. You can concatenate the parts by hand, removing the mail header, and then use 'uudecode' to decode them. We suggest however that you also get 'uud.c', which skips the mail headers automatically and is also able to fix up transmission errors caused by 'EBCDIC' machines. You can also get single parts of a file by sending 'get GAP '. For users in the United Kingdom with only Janet access, neither 'ftp' nor the mail server will work (please do *not* try to use the mail server). Please contact Derek Holt (e-mail address 'dfh@maths.warwick.ac.uk'). He has kindly offered us to distribute GAP in the United Kingdom. The 'ftp' directory and the 'listserv' archive contain the following files. Please check first which files you need, to avoid transferring those that you don't need. 'README': the file you are currently reading. GAP version 3 release 1 itself comes in several files. You do not need all of those files. All files are 'compress'-ed 'tar' archives. 'src3r1.tar.Z': the *source code* for the GAP kernel. You need this unless you get one of the executables below. This file is about 600 KBytes long. 'lib3r1.tar.Z': the *library of functions*. You need this. This file is about 700 KBytes long. 'doc3r1.tar.Z': the *documentation*. Serves as LaTeX source for the printed manual and online documentation. Contains further installation information. This file is about 700 KBytes long. 'doc3r1.dvi.Z': the preformatted documentation. You need this if you do not have a *big* TeX. This file is about 900 KByte long. 'grp3r1.tar.Z': various *group libraries*. Contains for example all primitive permutation groups of degree at most 50. This file is about 50 KByte long. 'two3r1.tar.Z': the library of *2-group* of size at most 256. This file is about 630 KByte long. 'tbl3r1.tar.Z': a library of *character tables* including all of the ATLAS. This file is about 2000 KByte long. 'src3r1.zoo', 'lib3r1.zoo', 'doc3r1.zoo', 'grp3r1.zoo', 'tbl3r1.zoo', 'two3r1.zoo': 'zoo' archives containing *exactly* the same files as the 'compress'-ed 'tar' archives above. The advantage of 'compress'-ed 'tar' archives is that 'uncompress' and 'tar' are widely available on UNIX systems. The advantage of 'zoo' archives is that they are smaller (about 30 percent) and that 'zoo' is more common on PC-s and Atari ST-s. (These files may not be available on all servers) We supply executables for some of the more popular machines. If you have one of those machines it will be easier for you to get this executable instead of compiling GAP yourself. The following executables are available (again these files may not be available on all servers) 'gapexe.st': executable for Atari ST (680?0) running TOS compiled with the GNU C compiler. This file is about 360 KByte long. 'gapexe.su3': executable for SUN 3 (680?0) running SunOS 4.0 or higher compiled with SunOS C. This file is about 410 KByte long. 'gapexe.su4': executable for SUN 4 (Sparc) running SunOS 4.0 or higher compiled with SunOS C. This file is about 450 KByte long. 'gapexe.dec': executable for DECstation (MIPS) running Ultrix 4.0 or higher compiled with Ultrix C. This file is about 500 KByte long. 'gapexe.next': executable for the NeXT (680?0) running MACH compiled with GNU C compiler. This file fixes a small problem in the source, so you should get this file instead of compiling the source yourself. This file is about 300 KByte long. To incorporate the latest changes into GAP 3.1, you also need the following *upgrade* files. Note that all the above files are those of the base version V3R1. Thus you *must* apply these upgrades to bring GAP to the latest patchlevel. The upgrade files are (unified) 'diff' files, so the upgrades can be made automatically with the program 'patch'. The upgrade files contain instructions how this can be done. 'upg3r1p1.dif.Z': This file contains the upgrade from base version (V3R1) to version 3 release 1 patchlevel 1 (V3R1P1). In this upgrade nothing important changed in the kernel, so you can use the above executables. This file is about 45 KBytes long. The following support files are also available (and again these files may not be available on all servers) 'compress.tar': 'compress' version 4.1. You need this program to uncompress the compressed tar files. Note however, that almost all UNIX systems these days already come with an executable 'compress'. This file is about 90 KByte long. 'patch.tar.Z': Larry Wall's 'patch' program version 2.0.2.0 (patchlevel 12u4). This program can be used to automatically apply upgrades. Note that older versions of 'patch' are *not* able to understand the unified 'diff' format used in the upgrade files. This file is about 70 KByte long. 'uud.c': 'uud' version 3.4. 'uud' is much better than the 'uudecode' that comes with most UNIX systems. This file is about 12 KByte long. 'zoo21.tar.Z': Rahul Dhesi's 'zoo' archiver version 2.1. You need this to unpack the *zoo-archives*. Note that the widespread version 2.01 will *not* work. This file is about 250 KByte long. 'zooexe.st': Executable of 'zoo' for the Atari ST. This file is about 80 KByte long. How to install GAP ================== The file 'install.tex' in 'doc3r1.tar.Z' contains extensive installation instructions. If however, you are one of those who never read manuals, here is a quick installation guide. Make a directory for GAP, e.g., '~/gap/' or '/usr/local/lib/gap/'. Unpack the source archive 'src3r1.tar.Z' into the subdirectory 'src/'; unpack the library archive 'lib3r1.tar.Z' into the subdirectory 'lib/'; unpack the documentation 'doc3r1.tar.Z' into the subdirectory 'doc/'. If you have obtained the optional groups and character tables libraries 'grp3r1.tar.Z', 'tbl3r1.tar.Z', and 'two3r1.tar.Z', unpack them into the subdirectories 'grp/', 'tbl/', and 'two/'. Change into 'src/' and execute 'make' to see a list of possible targets; select a target, if in doubt use 'bsd' or 'usg', and make the kernel. In an appropriate directory, e.g., '~/bin/' or '/usr/local/bin/', create a shell script that executes the GAP kernel. This should look like exec /src/gap -m 4m -l /lib/ $* The option '-m' specifies the amount of initial memory; the option '-l' specifies where to find the library, if you get it wrong GAP complains gap: hmm, I cannot find 'lib/init.g', maybe use option '-l '? Change into 'doc/' and make the printed manual with the commands latex manual; latex manual; lp -dvi manual.dvi or something similar, according to your local custom for using LaTeX. Try something in GAP, e.g., the following exercises GAP quite a bit gap> m11 := Group( (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) );; gap> Number( ConjugacyClasses( m11 ) ); The result should be 10. The Future of GAP ================= See ye not all these things? Verily I say unto you, there shall not be left here one stone upon another, that shall not be thrown down. (Matthew 24:2) Clearly GAP will contain bugs, as any system of this size, though currently we know none. Also there are things that we feel are still missing, and that we would like to include into GAP. We will continue to improve and extend GAP. We will release new versions quite regulary now, about three or four upgrades a year are planned. We are committed however, to staying upward compatible from now on in future releases. That means that everything that works now will also work in those future releases. This is different from the quite radical step from GAP 2.4 to GAP 3.1, in which almost everything was changed. Of course, we have ideas about what we want to have in future versions of GAP. However we are also looking forward to your comments or suggestions. The GAP Forum ============= We have also established a GAP forum, where interested users can discuss GAP related topics by e-mail. In particular this forum is for questions about GAP, general comments, bug reports, and maybe bug fixes. We, the developers of GAP, will read this forum and answer questions and comments, and distribute bug fixes. Of course others are also invited to answer questions, etc. We will also announce future releases of GAP on this forum. So in order to be informed about bugs and their fixes as well as about additions to GAP we recommend that you subscribe to the GAP forum. To subscribe send a message to 'listserv@samson.math.rwth-aachen.de' containing the line 'subscribe gap-forum ', where should be your full name, not your e-mail address. You will receive an acknowledgement, and from then on all e-mail messages sent to 'gap-forum@samson.math.rwth-aachen.de'. 'listserv@samson.math.rwth-aachen.de' also accepts the following requests: 'help' for a short help on how to use 'listserv', 'unsubscribe gap-forum' to unsubscribe again, 'recipients gap-forum' to get a list of subscribers, and 'statistics gap-forum' to see how many e-mail messages each subscriber has sent so far. If you have further questions or comments do not hesitate to write to me 'Martin.Schoenert@Math.RWTH-Aachen.DE'. Thank you for your attention, Martin. -- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany