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# Circle

## Adjoint groups of finite radical algebras

Version 1.1

August 2006

Alexander Konovalov
e-mail: konovalov@member.ams.org
WWW: http://homepages.vub.ac.be/~okonoval/
Address:
Department of Mathematics
Zaporozhye National University
Zaporozhye, 69063 Ukraine
Current address:
Department of Mathematics
Vrije Universiteit Brussel
Pleinlaan 2, Brussels
B-1050 Belgium

Panagiotis Soules
e-mail: psoules@math.uoa.gr
Address:
Department of Mathematics
National and Capodistrian University of Athens
Panepistimioupolis, GR-15784, Athens, Greece

### Abstract

The GAP4 package Circle extends the GAP functionality for computations in adjoint groups of associative rings. It provides functionality to construct circle objects that will respect the circle multiplication r * s = r + s + rs, create multiplicative groups, generated by such objects, and compute groups of elements, invertible with respect to this operation, for finite radical algebras and finite associative rings without one.

### Copyright

(C) 2006 by Alexander Konovalov and Panagiotis Soules

Circle is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site http://www.gnu.org/licenses/gpl.html.

If you obtained Circle, we would be grateful for a short notification sent to one of the authors.

If you publish a result which was partially obtained with the usage of Circle, please cite it in the following form:

A. Konovalov, P. Soules. Circle --- Adjoint groups of associative rings, Version 1.1; 2006 (http://homepages.vub.ac.be/~okonoval/circle.htm).

### Acknowledgements

We acknowledge very much Alexander Hulpke for helpful comments and advise.

### Contents

1. Introduction
1.1 General aims
1.2 Installation and system requirements
2. Circle functions
2.1 Circle objects
2.1-1 CircleObject
2.1-2 UnderlyingRingElement
2.1-3 IsCircleObject
2.1-4 IsPositionalObjectOneSlotRep
2.1-5 CircleFamily
2.2 Operations with circle objects
2.2-1 OneOp
2.2-2 InverseOp
2.2-3 IsUnit
2.2-4 IsCircleUnit
2.3 Construction of the adjoint group
2.3-1 AdjointGroup
2.4 Service functions
2.4-1 InfoCircle
2.4-2 CIRCLEBuildManual
2.4-3 CIRCLEBuildManualHTML
3. A sample computation with Circle

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