Group theory

Group theorythat branchmathematics concerned withstudygroups.

Please refer toGlossarygroup theory fordefinitionsterms used throughout group theory.

See also listgroup theory topics.

Tablecontents

1 History 2 Elementary introduction 3 Some useful theorems 4 Generalizations 5 Miscellany 6 External link

History

Therethree historial rootsgroup theory: theoryalgebraic equations, number theorygeometry. Euler, Gauss, Lagrange, AbelGalois were precedent researchers infieldgroup theory. Galoishonored asfirst mathematician linking group theoryfield theory, whose theorynow called Galois theory.

It was Walter Van Dyck who1882 gavemodern definition ofgroup.

Other important mathematiciansthis subject area includes Artin, Noether, Sylow,many more.

Elementary introduction

Groupsused throughout mathematics andsciences, oftencaptureinternal symmetryother structures, informautomorphism groups.

In Galois theory, which ishistorical origin ofgroup concept, one uses groupsdescribesymmetries ofequations satisfied bysolutions topolynomial equation. The solvable groupsso-named becausetheir prominent rolethis theory.

Abelian groups underlie several other structures thatstudiedabstract algebra, such as rings, fields,modules.

In algebraic topology, groupsuseddescribe invariantstopological spaces (the name oftorsion subgroupan infinite group showslegacythis fieldendeavor). Theycalled "invariants" because theydefinedsuchway thatdon't change ifspacesubjectedsome deformation. Examples includefundamental group, homology groupscohomology groups.

The conceptLie group (namedmathematician Sophus Lie)important instudydifferential equationsmanifolds;combine analysisgroup theory andthereforeproper objectsdescribing symmetriesanalytical structures. Analysis on theseother groupscalled harmonic analysis.

In combinatorics,notionpermutation group andconceptgroup actionoften usedsimplifycounting ofsetobjects; seeparticular Burnside's lemma.

An understandinggroup theoryalso important inphysical sciences. In chemistry, groupsusedclassify crystal structures, regular polyhedra, andsymmetriesmolecules. In physics, groupsimportant becausedescribesymmetries whichlawphysics seemobey. Physicistsvery interestedgroup representations, especiallyLie groups, since these representations often pointway to"possible" physical theories.

Some useful theorems

• Some basic resultselementary group theory
• Butterfly lemma
• Fundamental theorem on homomorphisms
• Jordan-Hölder theorem
• Krull-Schmidt theorem
• Lagrange's theorem
• Sylow theorems

Generalizations

In abstract algebra, we get some related structures whichsimilargroups by relaxing some ofaxioms given attop ofarticle.

• If we eliminaterequirement that every element have an inverse, then we getmonoid.
• If we additionally do not require an identity either, then we getsemigroup.
• Alternatively, if we relaxrequirement thatoperation be associative while still requiringpossibilitydivision, then we getloop.
• If we additionally do not require an identity, then we getquasigroup.
• If we don't require any axioms ofbinary operation at all, then we getmagma.

Groupoids, whichsimilargroups except thatcomposition a * b need not be definedall ab, arise instudymore involved kindssymmetries, oftentopologicalanalytical structures. Theyspecial sortscategories.

Lie groups, algebraic groupstopological groupsexamplesgroup objects: group-like structures sitting incategory other thanordinary categorysets.

Abelian groups formprototype forconceptan abelian category, which has applicationsvector spacesbeyond.

Formal group lawscertain formal power series which have properties much likegroup operation.

Miscellany

James Newman summarized group theory as follows:

The theorygroups isbranchmathematicswhich one does somethingsomethingthen comparesresults withresultdoingsame thingsomething else, or something else tosame thing.

External link

• History ofabstract group concept