Finite group tables in APL

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Finite group tables in APL

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International Conference on APL archive
Proceedings of the 2003 conference on APL: stretching the mind table of contents

San Diego, California

Pages: 75 - 81

Year of Publication: 2003

ISBN:1-58113-668-4

Authors

Samir Lipovaca	New York Stock Exchange, Inc., New York NY
Joseph Burchfield	New York Stock Exchange, Inc., New York NY

Sponsor

SIGAPL: ACM Special Interest Group on APL Programming Language

Publisher

ACM New York, NY, USA

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ABSTRACT

It is known from group theory that multiplication tables for finite groups are Latin squares. These basic Latin squares have proved useful for solving various problems, for example, problems of memory management in parallel computing systems, organizing party games, communications networking, computer imaging, and experimental design. We show how the problem of multiplication tables for a finite group G of order N can be solved using APL. The solution is based on Lagrange's and Cayley's theorems for finite groups: the order of a subgroup of a finite group G is a divisor of the order of the group and every group G of order N is isomorphic with a subgroup of the group of permutations of N objects (symmetric group SN). We demonstrate that using APL interactively offers a powerful, consistent, and simple notation for dealing with the elements of the symmetric group SN. Group tables for the groups of order 4, 6, and 8 are used to illustrate the method. Possible simplifications of the method are outlined.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	<u>http://www-groups.dcs.stand.ac.uk/~history/Mathematicians</u> (Galois and Cauchy html pages)
	2	Hamermesh, M., <u>Group theory and its application to physical problems</u>. Dover Publications, Inc., New York, 1989.
	3	Update Newsletter May 1990, Volume 2, Number 4, Published by The Center For Systems Science Simon Fraser University Burnaby, BC Canada V5A 1S6 604-291-3455 Editor: Barry Shell shell@cs.sfu.ca: <u>Partying Wtih A Latin Square</u>.
	4	Weatherburn, C. E. <u>A First Course in Mathematical Statistics</u> (Cambridge: C. U. P., 1968), Chapter XI.
	5	Burington, R. S. and May, D. C. <u>Handbook of Probability and Statistics with Tables</u> (Sandusky, Ohio: Handbook Publishers, Inc., 1958), pp. 276--279.
	6	Robert Mandl, Orthogonal Latin squares: an application of experiment design to compiler testing, Communications of the ACM, v.28 n.10, p.1054-1058, Oct. 1985 [doi>10.1145/4372.4375]