有限Abel群的结构定理（Fundamental Theorem ofFinite Abelian Groups）有限Abel群的结构定理(Fundamental Theorem of Finite Abelian Groups) 有限Abel群是群论中已被研究清楚了的重要群类，也是应用比较广泛的群类，本节的主要结论是有限Abel群可以分解成阶为素数的方幂的循环群(循环p-群)的直积，而且表法是唯一的。
我们先看几个具体的例子。
4阶群都是Abel群，它们有两种互不同构的类型，代表分别是。
Z,Z，Z422 ，其中是非Abel群;是Abel群，且6阶群有两种不同的类型，代表分别是ZZ,SS6633。
Z,Z，Z6238阶Abel群有三种不同的类型，代表分别是。
Z,Z，Z,Z，Z，Z8242229阶群都是Abel群，它们有两种互不同构的类型，代表分别是。
Z,Z，Z933 这些有限Abel群都同构于循环群或者循环群的直积，并且每个循环群的阶都是一个素数的方幂，这些循环群的阶组成的有重集合正好是该群阶素数方幂乘积的所有可能组合。
例如8阶32Abel群，有三种情形:，分别对应于8写成素数方幂乘积所有可能的形式{2},{2,2},{2,2,2}32(三种):。
8,2,8,2，2,8,2，2，2下面我们讨论一般有限Abel群的结构。
引理1 设a是群G的一个元素，a的阶等于。
其中与是两个互素的正整数，m,mmmm1212那么a可以唯一的表示成，式中的阶是;;而且都am(i,1,2)a(i,1,2)a,aaaa,aaii12i1221是a的方幂。
证明因为与互素，所以存在整数使得。
于是mmu,uum，um,112121122umumum，umumumumum2211112211222211，令，则，而且a,a,a,aa,aa,aaa,a,aa,aa121221mm12都是的方幂。
因为，所以的阶是的因子。
由于a(i,1,2)adm(i,1,2)ma,e,a,eaiiii112与互素，从而互素，并且，故的阶等于。
离散数学英中名词对照表英文Abel categoryAbel group (commutative group) Abel semigroup Abelian groupabsorption property accessibility relation acyclicaddition principleadequate set of connectives adjacentadjacent matrixadjugateadjunctionaffine planealgebraic closed field algebraic element algebraic extensionalphabetalternating groupannihilatorantecedentanti symmetryanti-isomorphismarc setargumentarityarrangement problem associateassociativeassociative algebraassociatorasymmetricatomatomic formulaaugmenting pigeon hole principle augmenting path automorphism automorphism group of graph auxiliary symbol A 离散数学英文—中文名词axiom of choiceaxiom of equalityaxiom of extensionalityaxiom of infinityaxiom of pairsaxiom of regularityaxiom of replacement for the formulaaxiom of the empty setaxiom of unionB balanced imcomplete block designbarber paradoxbase (base 2 exponential function)base (logarithm function to the base 2)Bell numberBernoulli numberBerry paradoxbiconditionalbijection (one-to-one correspondence)bi-mdulebinary relationbinary operationbinary symmetric channel (BSC)binary treebinomial coefficientbinomial theorembinomial transform bipartite graphblockblockblock codeblock designBondy theoremBoolean algebra Boolean expression Boolean functionBoole homomorophism Boole latticeBoolean matrixBoolean productbound occurrencebound variablebounded latticeBruijn theorem Burnside lemmaC cagecancellation property canonical epimorphism Cantor conjecture Cantor diagonal method Cantor paradoxcapacitycardinal number cardinalityCartesion product of graph Catalan numbercatenationCayley graphCayley theoremceiling functioncell (block)centercertain eventchain (walk) characteristic function characteristic of ring characteristic polynomial check digitsChinese postman problem chromatic number chromatic polynomial circuitcirculant graph circumferenceclassclassical completeness classical consistent cliqueclique numberclose with respect to closed termclosureclosure of graphcode elementcode lengthcode wordcoefficientcoimageco-kernalcoloringcoloring problemcombinationcombination numbercombination with repetationcommon divisorcommon factorcommutativecommutative diagramcommutative ringcommutative seimgroupcomparablecompatible withcomplementcomplement elementcomplement of B with respect to A complementary relation complemented latticecomplete bipartite graphcomplete graphcomplete k-partite graphcomplete latticecomplete matchcomplete n-treecompositecomposite operationcomposition (molecular proposition) composition of graph (lexicographic product) compound statementconcatenation (juxtaposition) concatenation graphconditional statement (implication) congruence relationcongruent toconjectureconjunctive normal form connected component connective connectivityconnectivity relation consecutively consequence (conclusion) conservation of flow consistent (non-contradiction) constructive proofcontain (in)contingencycontinuumcontraction of graph contradiction contravariant functor contrapositiveconversecoproductcorankcorresponding universal map countable (uncountable) countably infinite set counter examplecountingcovariant functorcoveringcovering numbercrossing number of graph cosetcotreecutcut edgecut vertexcyclecycle basiscycle matrixcycle rankcycle spacecycle vectorcyclic groupcyclic indexcyclic permutation cyclic semigroupD De Morgan's law decision procedure decoding table deduction theorem degreedegree sequence derivation algebra Descartes product descendant designated truth value deterministic diagonal functor diagonal matrix diameterdigraphdilemmadirect consequence direct limitdirect sumdirected by inclutiondisconnecteddiscrete Fourier transform discrete graph (null graph) disjoint setdisjunctiondisjunctive normal form disjunctive syllogism distancedistance transitive graph distinguished element distributivedistributive lattice divisibilitydivision subringdivison ringdivisor (factor) dodecahedrondomaindual categorydual formdual graphdual principledual statementdummy variableE eccentricityedge chromatic number edge coloringedge connectivityedge coveringedge covering numberedge cutedge setedge-independence number eigenvalue of graph element (entry) elementary divisor ideal elementary product elementary sumempty graphempty relationempty set endomorphismendpointentry (element) enumeration function epimorphismequipotentequivalenceequivalent category equivalent class equivalent matrix equivalent object equivalent relationerror functionerror patternEuclid algorithmEuclid domainEuler characteristicEuler circuitEuler functionEuler graphEuler numberEuler pathEuler polyhedron formula Euler tourEuler traileven permutationeventeverywhere defined excess capacity existence proof existential generalization existential quantification existential quantifier existential specification explicitextended Fibonacci number extended Lucas number extensionextension field extension graphexterior algebraF facefactorfactorablefactotialfactorizationfaithful (full) functor Ferrers graphFibonacci numberfieldfilterfinite dimensional associative division algebra finite extensionfinite field (Galois field )finite groupfinite setfinitely generated modulefirst order theory with equalityfive-color theoremfive-time-repetitionfixed pointfloor functionflowforestforgetful functorfour-color theorem (conjecture)F-reduced productfree elementfree monoidfree occurrencefree R-modulefree variablefree-Ω-algebrafull n-treefunction schemeG Galileo paradoxGauss coefficientGBN (G?del-Bernays-von Neumann system) GCD (Greatest Common Divisor) generalized Petersen graphgenerating functiongenerating proceduregeneratorgenerator matrixgeneric elementgenusgirthG?del completeness theoremgolden section numbergraceful graphgraceful tree conjecturegraphgraph of first class for edge coloring graph of second class for edge coloring graph rankgraph sequencegreatest common factorgreatest elementgreedy algorithmGrelling paradoxGr?tzsch graphgroupgroup codegroup of graphgrowth of functionHajós conjectureHamilton cycleHamilton graphHamilton pathHarary graphhash functionHasse diagramHeawood graphheightHerschel graphhom functorhomemorphism homomorphism homomorphism image homomorphism of graph hyperoctahedronhypothelical syllogism hypothesis (premise)idealidempotentidentityidentity functionidentity natural transformation imageimbeddingimmediate predcessor immediate successorimpossible eventincidentincident axiomincident matrixinclusion and exclusion principle inclusion relationindegreeindependentindependent number independent setindependent transcendental element indexindirected method H Iindividual variableinduced subgraphinfinite extensioninfinite groupinfinite setinitial endpointinitial objectinjectioninjection functorinjective (one to one mapping) inner faceinner neighbour setinorder searchintegral domainintegral subdomaininternal direct sum intersectionintersection of graph intersection operation intervalinvariant factorinvariant factor idealinverseinverse limitinverse morphisminverse natural transformation inverse operationinverse relationinversioninvertableinvolution property irreflexiveisolated vertexisomorphic categoryisomorphismisomorphism of graphjoinjoin of graphJ Jordan algebraJordan product (anti-commutator)Jordan sieve formulaj-skewjuxtapositionk-chromatic graphk-connected graphk-critical graphk-edge chromatic graphk-edge-connected graphk-edge-critical graph Kanaugh mapkernelKirkman schoolgirl problem Klein 4 groupKonisberge Brudge problem Kruskal's algorithm Kuratowski theoremlabeled graphLah numberLatin rectangleLatin squarelatticelattice homomorphismlawLCM (Least Common Multiple) leader cosetleast elementleafleast upper boundleft (right) identityleft (right) invertible element left (right) moduleleft (right) zeroleft (right) zero divisorleft adjoint functorleft cancellableleft cosetlengthlexicographic orderlLie algebraline- grouplinear array (list)linear graphlinear order (total order)K Llinear order set (chain)logical connective logical followlogically equivanlent logically implies logically valid loopLucas numbermagicmany valued proposition logic map coloring problem matchingmathematical structure matrix representation maximal element maximal idealmaximal outerplanar graph maximal planar graph maximum flow maximum matching maxtermmaxterm normal form (conjunctive normal form)McGee graph meetMenger theorem Meredith graph message word mini term minimal -connected graph minimal polynomial minimal spanning tree Minimanoff paradox minimum distance Minkowski summinterm (fundamental conjunctive form)minterm normal form (disjunctive normal form)M?bius function M?bius ladder M?bius transform (inversion)modal logic modelmodule homomorphismMkmoduler latticemodulusmodus ponensmodus tollensmodule isomorphismmonic morphismmonoidmonomorphismmorphism (arrow)M?bius functionM?bius ladderM?bius transform (inversion)multigraphmultinomial coefficientmultinomial expansion theoremmultiple-error-correcting codemultiplication principlemutually exclusivemultiplication tablemutually orthogonal Latin squareN n-ary operationn-ary productn-ary tree (n-tree)n-tuplenatural deduction systemnatural homomorphismnatural isomorphismnatural transformationnearest neighbernegationneighbour setnext state transition functionnon-associative algebranon-standard logicNorlund formulanormal formnormal modelnormal subgroup (invariant subgroup)n-relationnull graph (discrete graph)null objectnullary operationobjectodd permutationoffspringone to oneone-to-one correspondence (bijection) onto optimal solutionorbitorderorder (lower order,same order) order ideal order relationordered pairOre conditionorientationorthogonal Latin square orthogonal layoutoutarcoutdegreeouter faceouter neighbourouterneighbour setouterplanar graphpancycle graphparallelismparallelism classparentparity-check codeparity-check equationparity-check machineparity-check matrixpartial functionpartial ordering (partial relation) partial order relation partial order set (poset)partitionpartition number of integerpartition number of setPascal formulapathperfect code O Pperfect t-error-correcting code perfect graph permutationpermutation grouppermutation with repetation Petersen graphp-graphPierce arrowpigeonhole principleplanar graphplane graphPolish formPólya theorempolynomailpolynomial codepolynomial representation polynomial ring positional treepossible worldpostorder searchpower functorpower of graphpower setpredicateprenex normal formpreorder searchpre-ordered setprimary cycle modulePRIM's algorithmprimeprime fieldprime to each otherprimitive connectiveprimitive elementprimitive polynomialprincipal idealprincipal ideal domainprinciple of dualityprinciple of mathematical induction principle of redundancy probabilisticprobability (theory)productproduct categoryproduct partial orderproduct-sum formproof (deduction)proof by contraditionproper coloringproper factorproper filterproper subgroupproperly inclusive relationproposition (statement)propositional constantpropositional formula (well-formed formula,wff) propositional functionpropositional variablepseudocodepullbackpushoutquantification theoryquantifierquasi order relationquaternionquotient (difference) algebraquotient algebraquotient field (field of fraction)quotient groupquotient modulequotient ring (difference ring , residue ring) quotient set Ramsey graph Ramsey number Ramsey theorem rangerankreachability reconstruction conjecture recursive redundant digits reflexiveregular expression regular graph R Qregular representationrelation matrixrelative setremainderreplacement theoremrepresentationrepresentation functorrestricted proposition formrestrictionretractionreverse Polish formRichard paradoxright adjoint functorright cancellableright factorright zero divisonringring of endomorphismring with unity elementR-linear independencerooted treeroot fieldrule of inferenceRussell paradoxS sample spacesatisfiablesaturatedscopesearchingsectionself-complement graphsemantical completenesssemantical consistentsemigroupseparable elementseparable extensionsequencesequentsequentialSheffer strokesiblingssimple algebraic extensionsimple cyclesimple extensionsimple graphsimple pathsimple proposition (atomic proposition) simple transcental extension simplicationsinkslopesmall categorysmallest element Socrates argument soundness (validity) theorem sourcespanning subgraph spanning treespectra of graphspetral radiussplitting fieldsquare matrixstandard modelstandard monomil statement (proposition) Steiner tripleStirling numberStirling transformstrong induction subalgebrasubcategorysubdirect product subdivison of graph subfieldsubformulasubdivision of graph subgraphsubgroupsub-modulesubmonoidsublatticesubrelationsubringsub-semigroup subscript。
阿贝尔群简单解释
阿贝尔群（Abelian group）是数学中的一个概念，它是一种特殊的群，其中每个元素的逆元是其自身。
换句话说，群中的每个元素都是其自身的逆元。
阿贝尔群在代数和拓扑学中都有重要的应用。
阿贝尔群的一个重要特性是它的所有元素都可以被分解为有限个元素的乘积，而且这些元素的逆元可以很容易地找到。
这个特性使得阿贝尔群在许多问题中可以更加容易地处理。
阿贝尔群的另一个重要特性是它的所有子群都是正规的。
这意味着，如果一个子群包含了群中的某个元素，那么它就包含了该元素的整个陪集。
这个特性使得阿贝尔群在研究群的结构时更加有用。
在拓扑学中，阿贝尔群的一个重要应用是处理基本群（fundamental group）。
基本群是用来描述一个拓扑空间中所有路径的等价类构成的群。
如果一个拓扑空间是阿贝尔的（即它的基本群是阿贝尔群），那么这个空间就有很多良好的性质。
例如，它的所有连通components都是开的，它的所有简单闭曲线都是互不相交的，等等。
阿贝尔群的概念和理论在代数学和拓扑学中都有广泛的应用。
