ATLAS: Words and programs
The following information is available:
Words for generators of subgroups, representatives of conjugacy,
etc. are given as `straight-line programs' to calculate the
corresponding group elements.
These programs are designed to be both machine-readable and
human-readable, as far as possible.
Commands
There are the following commands:
- inp
- input.
- oup
- output.
- mu
- multiply.
- iv
- invert.
- pwr
- power.
- cj
- conjugate.
- cjr
- conjugate and replace.
- com
- commutator.
Syntax
The syntax is as follows:
- [inp [n [i1 i2 ... in]]]
- input n generators (default 2), called i1 ... in (default 1 2 ... n).
- [oup [n [i1 i2 ... in]]]
- output n elements (default 2), called i1 ... in (default 1 2 ... n).
- mu a b c
- c := a * b
- iv a b
- b := a-1
- pwr n a b
- b := an, for n an integer bigger than 1.
- cj a b c
- c := b-1ab
- cjr a b
- a := b-1ab
- com a b c
- c := a-1b-1ab
All input lines must precede program lines, which must
precede all output lines. Only the first input or output line
may omit the names of the elements.
No command except cjr may overwrite its own input.
Filenames are generally of the form A-B, where A specifies the
required input, and B specifies the desired output.
Thus A is typically a group name followed by the name of
a specified generating set. For example M11G1 denotes the
`standard generators' of the group M11.
Similarly, B is typically a name describing the type of output,
followed by W and a version number. For example,
cyc denotes maximal cyclic subgroups, and ccls denotes
conjugacy classes of elements, while max5 denotes the a representative of
the fifth conjugacy class of maximal subgroups (ATLAS ordering).
Examples:
- M11G1-cycW1
- From standard generators of M11, make representatives
of the classes of maximal cyclic subgroups. This is the first
version of such a program (denoted by the W1 in the name).
- M11G1-cclsW1
- From standard generators of M11, make representatives
of all the conjugacy classes.
- L35G2-G1W1
- From G2 standard generators of L3(5), make the G1 standard generators.
- U62G1-max4W1
- From standard generators of U6(2), make a representative
of the fourth class of maximal subgroups. The generators of
the subgroup are not defined abstractly, but only by the
words in the program (W1).
- U62G1-max4W2
- From standard generators of U6(2), make a representative
of the fourth class of maximal subgroups. The generators of
the subgroup are not defined abstractly, but only by the
words in the program (W2). Note that the subgroup output does not
need to be equal to that which was output from U62G1-max4W1, but must
be conjugate to it.
- M11G1max2W1-L211G1W1
- From the output of the program M11G1-max2W1,
make generators of the subgroup which are (automorphic to)
standard generators of L2(11).
Note: The maximal subgroups of G.2 are numbered in decreasing order
of size (starting with G which is denoted max1), irrespective
of any possible corresponding ordering of the maximal subgroups of G.
The same principle applies to maximal subgroups of other G.A.
We extend the language described above in order to provide programs
to check various assertions, particularly things to do with standard
generators.
More details are given on the
black box algorithms page.
Programs available
We eventually hope to provide provide the following information:
(31/1/05): Please note that the naming convention for these programs
has recently changed.
- GroupG[n]-defn[m]
- A machine-readable definition of the specified generators of the specified group.
- GroupG[n]-find[m]
- A black-box algorithm for finding the specified generators of the specified group
(up to automorphisms).
- GroupG[n]-check[m]
- A black-box algorithm for checking that the generators are correct,
given that the group is correct.
- GroupGn[n]-prove[m]
- A procedure for proving that the generators are the correct
generators for the correct group. This will only be available for smallish groups.
Syntax of commands
Extra commands required for these programs include the following:
- chor a b
- Check that element a has order b
-
-
Examples
The file M11G1-check1 contains the following program, which check that
generator 1 has order 2, generator 2 has order 4,
their product ab has order 11, and the word ababbabbb has order 5.
chor 1 2
chor 2 4
mu 1 2 3
chor 3 11
mu 3 2 4
mu 3 4 5
mu 5 4 3
mu 3 2 4
chor 4 5
Go to main ATLAS (version 2.0) page.
Version 2.0 created on 2nd December 1999.
Last updated 31.01.05 by SJN.
R.A.Wilson, R.A.Parker and J.N.Bray.