ANU Polycyclic Quotient Programs
At the Mathematical Sciences
Institute of The Australian
National University (ANU), a suite of programs has been developed
for the computation of polycyclic factor groups of groups given by a
finite presentation. This page is intended to give a (very) brief
introduction to each program and provide links to sites from where
each program is available.
All three programs are available from the archive of the
Algebra Program at the Mathematical Sciences Institute. The
algorithms, details of the implementations and results obtained with
the programs are described in the list of
publications and their references.
The ANU p-Quotient Program
This program implements
- the p-quotient algorithm
- the p-group generation algorithm
- an algorithm to decide isomorphism of p-groups
- an algorithm to compute the automorphism group of a
p-group
It has been used to obtain results such as:
- there are 56092 groups of order 256
- the largest finite group generated by 3 elements
in which every element has order dividing 5 has order
52822.
- and (using variants for Lie and associative algebras
which are not yet in the public domain) that the
largest finite group generated by 2 elements in which
the order of every element divides 7 has order
dividing 720418.
The program is implemented in C and is available
as stand-alone or as part of the systems
GAP,
Magma
and Quotpic.
The ANU Nilpotent Quotient Program
The program implements
- a nilpotent quotient algorithm
- facilities for the computation of nilpotent groups
that satisfy an Engel identity or have Engel
elements as generators.
It has been used to
- obtain insight into the nature of right Engel elements
- compute polycyclic presentations for infinite
nilpotent Engel groups such as the free nilpotent
2-generator n-Engel group for n=4,5,6.
The implementation is written in C and is
available as stand-alone and as part of the systems
GAP,
Magma
and Quotpic.
The ANU Soluble Quotient Program
The program implements a finite soluble quotient
algorithm.
The program has been used to
- compute a polycyclic presentation for B(2,6), the
freest group on two generators with exponent 6.
- find small sets of sixth power relations that define
certain groups of exponent six.
The program is implemented in C and is available as
stand-alone, or as part of the system
GAP.
An alternative soluble quotient algorithm by Brückner
and Plesken is available in
Magma.
|