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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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.

Other web sites with this focus

Selected topics at this site

• How many groups of order n?
• How many groups of order p^n? [Derek Holt]
• Pyber's estimate for the number of groups of order n
• How many groups of order 2^n (n through 8)
• Table showing the number of groups of order N for all N < 200 except N=192.
• Where to find group tables for all the groups with order up to N .
• List of all groups of small order and an appeal to discount Cayley tables for their enumeration.
• Lagrange's theorem as an extension (to matrices) of Fermat's little theorem.
• Converse of Lagrange's Theorem implies solvable
• When every group of order n is abelian
• What are the possible orders of elements in symmetric groups? -- discussion
• What are the possible orders of elements in symmetric groups? -- literature review
• Maximum order of elements in the symmetric group S_n
• Todd and the odd number 6
• Constructing a finite group with three elements x, y, z having arbitrary orders and xyz=1.
• Groups isomorphic if equal counts of elements of each order? (No)
• G/Z(G) is a square if Z cyclic, contains G'
• Fixed-point-free automorphism of prime order implies nilpotent
• Groups with cyclic Sylow 2-subgroups have normal 2-complements
• General linear groups over a finite field suggest why every p-subgroup should lie in a Sylow-p-subgroup
• Computing a Sylow subgroup and its normalizer in GL(n,p)
• Groups in which distinct conjugacy classes have distinct sizes
• Groups which have a unique normal subgroup
• Characteristic subgroups and distinct isomorphic normal subgroups
• Finite groups with distinct but isomorphic characteristic subgroups
• How many simple groups of a bounded order?
• Structure of groups of order pq^2
• What are the sporadic simple groups?
• The Monster simple group, modular forms, "moonshine", and applications to physics.
• How to determine group size from the number of conjugacy classes.
• Will just a few numerical invariants characterize a group up to isomorphism? (no)
• Can we determine G from the cardinalities of all conjugacy classes (no)
• Announcement of p-group software.
• ANU p-quotient program (for p-groups)
• Pointer for the list of sporadic finite simple groups. (Other small simple groups too)
• All finite simple groups can be generated by two elements; indeed for alternating (and symmetric) groups, such pairs of generators are legion.
• Tabulating orbits under the action of the symmetric group.
• For what n is the symmetric group Sym(n) a Frobenius group?
• Applications of the symmetric group to change-ringing! (permuting the order of bells being rung).
• Connections between the Rubik's group and physics.
• Summary of information about generators and relations of the Rubik group.
• Generators and relations for the Rubik group, with an introduction to the GAP program.
• Wordlength in the Rubik group, with a URL.
• How many positions in an n-dimensional, edge-length k Rubik's cuboid?
• URLs for solution of Rubik's cube and related themes.
• Finding enough invariants to distinguish non-isomorphic groups
• Finding all finite groups in which same order elements are in same conjugacy class. (Examples easy, proof hard!)