info.4

4.  The shell program "cohomology".

Usage: cohomology [-m1crngxjsh] gpname

(Options may be given in any order, and need not be separated.)
This program also calls the short shell program "execcmd".

Purpose: To run the programs listed in info.2 in the correct order, for the
         computation of Schur multipliers of finite permutation groups G,
         and of first and second cohomology groups of G on modules M defined
         over a field of prime order p. Additionally, for the computation of
         presentations of covering groups of G, or of extensions of  the module
         M by G.

The user has two pieces of work to do.

1) (S)He must provide the necessary input data in the appropriate format.
   All files related to computation within a group G with name "gpname" have
   names of the form gpname.'name', where 'name' specifies the file.
   PLEASE HEED THE WARNING IN 'info.0' ABOUT MAXIMAL LENGTH OF 'gpname' (8).
   At the start of the computation the situation should be as follows:
   a) gpname.inperm, which lists the generating permutations of G,
      must be present.
      None of these generators is allowed to be the identity;
   b) gpname.inmat, which lists the matrices for these generators
      giving their action on the module M, must be present unless -m is set
      (multiplier computation);
   c) gpname.tc, which contains a presentation for G on these generators may
      be present. If so, then it will be used when constructing the presentation
      of any extension of the multiplier or of the module M by G. The full
      presentation will be printed both on the terminal and in a file called
      gpname.ext.tc at the end of the computation.
   d) no other files with names of this form should be present, since they could
      be overwritten, or, worse, their existence could mislead the shell program
      and cause errors or crashes.
   There is a short shell program 'clrgrp' present which is called by

   clrgrp gpname

   and removes all files of form gpname.* except those of form gpname.in* and
   gpname.tc*. This is sufficient to avoid the dangers described in d).
   For advice about methods of creating the files in a) b) and c), and for
   information on their formats, see info.0.

2) If the index of the normalizer N(P) of a Sylow subgroup P of G has large
   index in G, then the computations are speeded up considerably by the
   introduction of a chain of subgroups between N(P) and G. These subgroups
   H(1), H(2), ... should be stored in files with names gpname.sg1, gpname.sg2,
   etc., and they should satisfy
   N(P) < H(1) < H(2) < ... < G. (Strict inclusion is necessary here!)
   After the shell program has computed G, P and N(P), it returns control
   temporarily to the user (unless option -n is set) for him to create these
   files. To do this effectively, some knowledge of the individual programs
   described in info.3 is necessary. 
   In particular, egrun with the -s flag, to create point stabilizers,
   normrun, to compute normalizers and centralizers, and optrun to compute
   other functions like intersections, commutator groups and cores are likely
   to be useful.

Meaning of Options.

Usage: cohomology [-m1crnxjsh] gpname
-m)  Compute the p-part of the Schur multiplier of G. The prime p is read from
     the terminal by "cohomology", at the beginning.

-1)  Compute the first cohomology group of G on the module M defined by the
     given matrices.

     -m) and -1) cannot both be set together, and if neither is set, then the
     default action is to compute the second cohomology group of G on M.

-c)  (Corestriction). After computing the multiplier or second cohomology group,
     compute a presentation for the covering group of G (if -m set) or of an
     extension of the module M by G. In the second case, if H^2(G,M) has 
     dimension > 1, then this extension is not unique, and the user has to
     select a particular generator. This is done by inputting the corresponding
     vector in H~2(G,M) at the terminal.
     At present, it is not possible to set -1 and -c together.

-r)  (Resume). This only makes sense if H^2(G,M) has already been computed
     in a previous run of "cohomology" with the -c option.
     It may not be called with -1 or -m, and -c need not be explicitly set,
     since it is assumed.
     The user can immediately select another generator of H^2(G,M) for
     the computation of the extension.

-n)  Do not return control to the user for the computation of intermediate
     subgroups. Use this if |G:N(P)| is known to be small.

-g)  Stop after group and matrix theoretic calculations; i.e. before the
     first call of nqrun or nqmrun.

-x)  Omit various checks that are normally carried out by "cohomology". They
     do not take much time, and they could help to avoid confusing faults, so
     it is not advisable to call this option.
     The checks are as follows:
     a) Check whether the subgroups H(1), H(2) really satisfy the required
        strict inclusion conditions;
     b) Check that the given matrices really do define a representation of the
        group G that is defined by the generating permutations. This is done
        by checking that the matrices satisfy the relators in a presentation of
        G.
     Other possible input errors should result in comprehensible diagnostics,
     although silly errors like 'p' not being prime may cause havoc.
     See info.0 for comments on errors caused by lack of space.

-j) Send standard output of all programs run to a file gpname.junk. Diagnostic
    output (including final results of the computation) will still appear on the
    terminal. Other output is only useful for observing the progress of the
    computation.

-h) Send a list of all programs run (excluding the checking programs) to a file
    gpname.history. This, and the following option, are useful if it is required
    to follow a computation more closely, repeat bits of it, or find out
    exactly where an error occurred.

-s) Operate in step-by-step mode. Before each command is executed, it is echoed
    on the terminal. The user can answer either 'n' (do not execute and
    proceed to next command), 'q' (quit the whole process), or 'y' (default -
    execute).
    The 'n' response should of course only be used if this command has already
    been executed on a previous run.