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# 20D: Abstract Finite Groups

## Introduction

In this section we discuss finite groups only. Moreover, we study the internal properties of those groups -- material on their representation (20C) or permutation actions (20B) or cohomology (20J) are for now on the main group theory page. One may characterize this page as holding all those results about group theory for which a consideration of the order of elements is a central part of the question.

We include here some topics in permutation groups when the groups are obviously finite (e.g. the Rubik group), although these will eventually be moved to a separate section 20B.

## History

Sylow. Frattini. Burnside. Brauer. Then the whole classification of finite simple groups. Après ça, la deluge.

## Applications and related fields

Up to parent page

## Subfields

• 20D05: Classification of simple and nonsolvable groups
• 20D06: Simple groups: alternating groups and groups of Lie type. See also 20Gxx, 22Exx
• 20D08: Simple groups: sporadic groups
• 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, pi-length, ranks, See also 20F17
• 20D15: Nilpotent groups, p-groups
• 20D20: Sylow subgroups, Sylow properties, pi-groups, pi-structure
• 20D25: Special subgroups (Frattini, Fitting, etc.)
• 20D30: Series and lattices of subgroups
• 20D35: Subnormal subgroups
• 20D40: Products of subgroups
• 20D45: Automorphisms
• 20D60: Arithmetic and combinatorial problems
• 20D99: None of the above but in this section

Parent field: 20: Group Theory and Generalizations

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Well-known textbooks

• Thorough graduate textbook series: B. Huppert, "Endliche Gruppen I", Springer, New York, 1967; N. Blackburn and B. Huppert, "Finite Groups II, III", Springer, Berlin, 1982. ISBN 3-540-10633-2 MR84i:20001a,b
• Suzuki, Michio "Group theory", Grundlehren 247-8 (2 volumes), Springer-Verlag, Berlin-New York, 1982. 434 pp. ISBN 3-540-10915-3 (MR82k:20001c) and 1986 621 pp. ISBN 0-387-10916-1 (MR87e:20001)
• A post-classification graduate text might be Aschbacher, Michael, "Finite group theory", Cambridge University Press, Cambridge-New York, 1986. 274 pp. ISBN 0-521-30341-9. MR89b:20001

"Reviews on finite groups", classified by Daniel Gorenstein. American Mathematical Society, Providence, 1974, 706 pp. -- reviews published 1940-1970 in Mathematical Reviews.

## Software and tables

• MeatAxe, calculation of modular character tables (or other matrix/finite field applications)
• SISYPHOS - Computing in modular group algebras of p-groups
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A.: "Atlas of finite groups: Maximal subgroups and ordinary characters for simple groups" Oxford University Press, Oxford, 1985, 252 pp. ISBN 0-19-853199-0. One-stop shopping for your tables of finite simple groups and related groups. For corrigienda and companion data see also the following item:
• Jansen, Christoph; Lux, Klaus; Parker, Richard; Wilson, Robert: "An atlas of Brauer characters", London Mathematical Society Monographs. New Series, 11. The Clarendon Press, Oxford University Press, New York, 1995, ISBN 0-19-851481-6
• Related material available through the web: the ATLAS of Finite Group Representations
• The Groups of Order 2^n (n <= 6), Marshall Hall Jr. and James K. Senior. The Macmillan Company, New York, 1964.

## Selected topics at this site

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Last modified 2000/01/14 by Dave Rusin. Mail: