Talk:Group theory

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Contents 1 Scope and structure 2 list of gp th topics 2.1 Basic properties of groups 2.1.1 Group homomorphisms 2.2 Basic types of groups 2.2.1 Simple groups and their classification 2.2.2 Permutation and symmetry groups 2.3 Concepts groups share with other mathematics 2.4 Mathematical objects making use of a group operation 2.5 Mathematical fields & topics making important use of group theory 2.6 Algebraic structures related to groups 2.7 Group representations 2.8 Computational group theory 2.9 Applications 2.10 Famous problems 2.11 Other topics 3 restructuring

[edit] Scope and structure

Now that Group (mathematics) has reached GA status, I feel that it is time to start working on this article and reduce overlap between the two articles. To do this I think we need to firmly establish what the scope of the article should be.

I feel that this article should mainly deal with the field of research in mathematics dealing with groups. So it should deal with the history of the subject (starting with Galois and abel probably), the major players (past and present), major results, open questions, and current branches of research. It should probably also say something about connections with others fields of research and thus applications. (note that these should be applications of the theory of groups-i.e. situations in which the results obtained by considering abstract groups are useful- not merely situations where groups are applied) (TimothyRias (talk) 08:05, 15 May 2008 (UTC))

Sounds good. I have a little trouble envisioning the result -- each major research area has its own history, players, results, and open questions, and similarly in each era, the division of group theory into subfields was different. It is probably a good idea to sketch out the table of contents here. JackSchmidt (talk) 12:33, 15 May 2008 (UTC)

Mock TOC:

• lead
• History

• History of the abstract group concept could be a decent source.
• Felix Klein's Erlangen program

• Major results

• Classification of finite simple groups, etc.

• Groups vs. geometry. Groups vs. symmetry
• Branches of group theory

• Representation theory
• Combinatorical group theory
• Geometric group theory

• Open problems/current branches of research.

• Open problems in combinatorical and geometric gp. theory

• Applications

• Applications in other fields of mathematics:

• (algebraic) topology: fundamental group, classifying space (BGLn) vs. group cohomology, K-theory
• number theory
• Algebraic geometry (abelian varieties, Jacobian varieties)
• Galois theory. Mention profinite groups. Also Differential Galois theory (corresponding to Lie groups)
• geometry via geometric group theory, geometric invariant theory

• Applications in other fields:

• cryptography (e.g. using ab. varieties over finite fields, in particular elliptic curves, Diffie-Hellman, fast exponentiation)
• coding theory
• Physics
• Chemistry (there are whole books about, e.g. Group Theory and Chemistry by Bishop)

Feel free to mess around with this.(TimothyRias (talk) 12:46, 15 May 2008 (UTC))

I left a comment at Talk:Group (mathematics). In short that article comments on what is a group, whereas here the focus is on why, how, when, where to. The suggested table of contents is a good step in this direction. Geometry guy 20:33, 15 May 2008 (UTC)

Fleshed out the TOC a little bit. (TimothyRias (talk) 15:15, 16 May 2008 (UTC))

Added some items to the TOC. It's getting longer and longer ... Jakob.scholbach (talk) 15:39, 16 May 2008 (UTC)

The List of group theory topics has a fairly good (and long) list of what could show up here. Jakob.scholbach (talk) 12:15, 18 May 2008 (UTC)

[edit] list of gp th topics

I copy below the content of the list of gp th topics. Items already mentioned or very close to ones covered are striked out. Indented items seem (to me) to be less relevant or irrelevant to the article. Jakob.scholbach (talk) 20:51, 2 June 2008 (UTC)

Central extension Direct product Direct sum Direct sum of groups Extension problem Free abelian group Free group Free product

Generating set of a group Group cohomology Group extension Presentation of a group Product of group subsets

Schur multiplier Semidirect product Sylow theorems Hall subgroup Wreath product

[edit] Basic properties of groups

Butterfly lemma Center of a group Centralizer and normalizer Characteristic subgroup Commutator Composition series Conjugacy class Conjugate closure Conjugation of isometries in Euclidean space

Core (group) Coset Derived group Elementary group theory Euler's theorem Fitting subgroup Generalized Fitting subgroup Hamiltonian group Identity element

Lagrange's theorem Multiplicative inverse Normal subgroup p-core Schreier refinement theorem Subgroup Transversal Torsion subgroup Zassenhaus lemma

[edit] Group homomorphisms

Automorphism Automorphism group Factor group Fundamental theorem on homomorphisms Group homomorphism Group isomorphism

Homomorphism Isomorphism theorem Inner automorphism Order automorphism Outer automorphism group Quotient group

[edit] Basic types of groups

Examples of groups Abelian group Cyclic group Rank of an abelian group Dicyclic group Dihedral group Divisible group Finitely generated abelian group

Group representation Klein four-group List of small groups Locally cyclic group Nilpotent group Solvable group P-group Pro-finite group

[edit] Simple groups and their classification

• Classification of finite simple groups

Algebraic group Alternating group Borel subgroup Chevalley group Conway group Feit-Thompson theorem Fischer group General linear group Group of Lie type

Group scheme Janko group Lie group Simple Lie group Mathieu group Monster group Baby Monster group Bimonster Parabolic subgroup

Projective group Simple group Quasisimple group Special linear group Symmetric group Thompson group (finite) Tits group Weyl group

[edit] Permutation and symmetry groups

Arithmetic group Braid group Burnside's lemma Cayley's Theorem Coxeter group Crystallographic group Crystallographic point group, Schoenflies notation Discrete group Euclidean group Even and odd permutations

Frieze group Frobenius group Fuchsian group Geometric group theory Group action Homogeneous space Isometry group Orbit (group theory) Permutation Permutation group

Rubik's Cube Space group Stabilizer subgroup Steiner system Strong generating set Symmetry Symmetric group Symmetry group Wallpaper group

[edit] Concepts groups share with other mathematics

Associativity Bijection Bilinear operator Binary operation Commutative

Congruence relation Equivalence class Equivalence relation Lattice (group) Lattice (discrete subgroup) Multiplication table Prime number Up to

[edit] Mathematical objects making use of a group operation

Abelian variety Algebraic group Banach-Tarski paradox Category of groups Dimensional analysis Elliptic curve Galois group

Gell-Mann matrices Group object Hilbert space Integer Lie group Matrix Modular arithmetic

Number Pauli matrices Real number Quaternion Quaternion group Tensor

[edit] Mathematical fields & topics making important use of group theory

Algebraic geometry Algebraic topology Galois theory Discrete space Fundamental group

Geometry Homology Minkowski's theorem Topological group

[edit] Algebraic structures related to groups

Field Finite field Group with operators Heap Linear algebra Magma Module

Monoid Monoid ring Quandle Quasigroup Ring Semigroup Vector space

[edit] Group representations

See also list of harmonic analysis and representation theory topics

Affine representation Character theory Great orthogonality theorem Maschke's theorem

Monstrous moonshine Projective representation Representation theory Schur's lemma

[edit] Computational group theory

• Coset enumeration
• Schreier's subgroup lemma
• Schreier-Sims algorithm
• Todd-Coxeter algorithm

[edit] Applications

Computer algebra system Cryptography Discrete logarithm Triple DES Caesar cipher

Exponentiating by squaring Knapsack problem Shor's algorithm Standard Model

[edit] Famous problems

• Burnside's problem
• Classification of finite simple groups
• Herzog-Schönheim conjecture
• Subset sum problem
• Whitehead problem
• Word problem for groups

[edit] Other topics

Amenable group Capable group Commensurability (mathematics) Compact group Compactly generated group Complete group Congruence subgroup

Continuous symmetry Frattini subgroup Growth rate Heisenberg group, discrete Heisenberg group Molecular symmetry Tarski monster group Thompson groups Transfer (group theory)

[edit] restructuring

I did some restructuring of the article involving mainly the lead and the history section. I am considering also removing the part giving the definition of group, this adequately dealt with in the group article and having this here adds to the critisims that the two articles have too much overlap. This article (at least in my view) should not be about groups but about the field that studies them. (TimothyRias (talk) 09:25, 22 May 2008 (UTC))

You are right, the lead was still totally out of shape. Probably this is the last we need to do, when everything else is pretty much done. History in front seems good to me. (I deleted the 2nd copy at the end. It also needs lots of improvement still, btw).

As for the overlap: as a matter of fact, groups and group theory (as corpora of certain knowledge) do overlap. My aim with the basic stuff section was to give a condensed intro to some definitions which are central to group theory and will be used later in the article (such as group (obviously ;)), subs, quotients, homomorphisms, normal and composition series etc.) Some of them could theoretically be eliminated because they are already in the group article. But I think it is much nicer for a reader to be reminded of what a normal subgp is when reading about normal series.

I would like to propose the following procedere: let's write up the article (as complete and comprehensive as possible), and then decide which content is moved where. Ideally, the gp article should be readable without prerequisites, the gp th article does have to refer to the knowledge of certain notions, but should nonetheless be somewhat self-contained. Jakob.scholbach (talk) 12:25, 22 May 2008 (UTC)

Agreed. The lead should probably be re-edited at the end. My current restructuring was mainly to make clear to new editors coming into the article what the difference between the article should be. (pre-empting any new merger/deletion proposals from editors wandering by noticing that the lead of group theory deals mainly with groups instead of the theory.

About the basic stuff. This should probably be dealt with on a case by case basis. I.e. introducing concepts when needed and referring to the relevant articles for a more indepth discussion. If we don't do this I feel the size of this article could quickly get out of hand. (TimothyRias (talk) 13:58, 22 May 2008 (UTC))