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| {{Group theory sidebar |Basics}}
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| In '''[[group theory]]''', a [[Group (mathematics)|group]] (''G'', •) is a [[Set (mathematics)|set]] ''G'' [[Closure (mathematics)|closed]] under a [[binary operation]] • satisfying the following 3 [[axioms]]:
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| * ''[[Associativity]]'': For all ''a'', ''b'' and ''c'' in ''G'', (''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'').
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| * ''[[Identity element]]'': There exists an ''e''∈''G'' such that for all ''a'' in ''G'', ''e'' • ''a'' = ''a'' • ''e'' = ''a''.
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| * ''[[Inverse element]]'': For each ''a'' in ''G'', there is an element ''b'' in ''G'' such that ''a'' • ''b'' = ''b'' • ''a'' = ''e'', where ''e'' is an identity element.
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| Basic examples for groups are the [[integers]] '''Z''' with addition operation, or [[rational numbers]] without zero '''Q'''\{0} with multiplication. More generally, for any [[ring (mathematics)|ring]] ''R'', the [[unit (ring theory)|units]] in ''R'' form a [[multiplicative group]]. See the [[group (mathematics)|group]] article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.
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| This glossary provides short explanations of some basic notions used throughout group theory. Please refer to [[group theory]] for a general description of the topic. See also [[list of group theory topics]].
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| ==Basic definitions==
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| A [[subset]] ''H'' ⊂ ''G'' is a ''[[subgroup]]'' if the restriction of • to ''H'' is a group operation on ''H''. It is called ''[[normal subgroup|normal]]'', if left and right [[coset]]s agree, i.e. ''gH'' = ''Hg'' for all ''g'' in ''G''. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup ''N'' in a group ''G'' naturally inherits a group structure, enabling the formation of the [[quotient group]], usually denoted ''G''/''N'' (also called a
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| ''factor group''). The ''[[butterfly lemma]]'' is a technical result on the [[lattice of subgroups]] of a group.
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| Given a subset ''S'' of a group ''G'', the smallest subgroup of ''G'' containing ''S'' is called the subgroup ''generated by S''. It is often denoted <''S''>.
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| Both subgroups and normal subgroups of a given group form a [[complete lattice]] under inclusion of subsets; this property and some related results are described by the [[lattice theorem]].
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| Given any set ''A'', one can define a group as the smallest group containing the [[free semigroup]] of ''A''. This group consists of the finite strings called words that can be composed by elements from ''A'' and their inverses. Multiplication of strings is defined by concatenation, for instance <math>(abb)*(bca)=abbbca.</math>
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| Every group ''G'' is basically a factor group of a free group generated by the set of its elements. This phenomenon is made formal with [[presentation of a group|group presentations]].
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| The ''[[direct product of groups|direct product]]'', ''[[direct sum of groups|direct sum]]'', and ''[[semidirect product]]'' of groups glue several groups together, in different ways. The direct product of a family of groups ''G''<sub>''i''</sub>, for example, is the [[cartesian product]] of the sets underlying the various ''G''<sub>''i''</sub>, and the group operation is performed component-wise.
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| A ''[[group homomorphism]]'' is a map {{nowrap|''f'' : ''G'' → ''H''}} between two groups that preserves the structure imposed by the operation, i.e.
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| :''f''(''a'' • ''b'') = ''f''(''a'') • ''f''(''b'').
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| [[Bijective]] ([[injective|in-]], [[surjective]]) maps are [[isomorphism]]s of groups ([[monomorphism|mono-]], [[epimorphism]]s, respectively). The [[kernel of a homomorphism|kernel]] ker(''f'') is always a normal subgroup of the group. For ''f'' as above, the ''[[fundamental theorem on homomorphisms]]'' relates the structure of ''G'' and ''H'', and of the kernel and image of the homomorphism, namely
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| :''G'' / ker(''f'') ≅ [[image (mathematics)|im]](''f'').
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| One of the fundamental problems of group theory is the ''classification of groups'' [[up to]] isomorphism.
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| Groups together with group homomorphisms form a [[category (mathematics)|category]].
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| In [[universal algebra]], groups are generally treated as algebraic structures of the form {{nowrap|(''G'', •, ''e'', <sup>−1</sup>)}}, i.e. the identity element ''e'' and the map that takes every element ''a'' of the group to its inverse ''a''<sup>−1</sup> are treated as integral parts of the formal definition of a group.
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| ==Finiteness conditions==
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| The ''[[order of a group|order]]'' |''G''| (or o(''G'')) of a group is the [[cardinality]] of ''G''. If the order |''G''| is (in-)finite, then ''G'' itself is called (in-)[[finite group|finite]]. An important class is the ''group of permutations'' or [[symmetric group]]s of ''N'' letters, denoted S<sub>''N''</sub>. ''[[Cayley's theorem]]'' exhibits any finite group ''G'' as a subgroup of the [[symmetric group]] on ''G''. The theory of finite groups is very rich. ''[[Lagrange's theorem (group theory)|Lagrange's theorem]]'' states that the order of any subgroup ''H'' of a finite group ''G'' divides the order of ''G''. A partial converse is given by the ''[[Sylow theorems]]'': if ''p''<sup>''n''</sup> is the greatest power of a [[prime number|prime]] ''p'' dividing the order of a finite group ''G'', then there exists a subgroup of order ''p''<sup>''n''</sup>, and the number of these subgroups is also known. A [[projective limit]] of finite groups is called [[profinite group|profinite]].<ref>{{harvnb|Shatz|1972}}</ref> An important profinite group, fundamental for [[p-adic analysis]], [[class field theory]], and [[l-adic cohomology]] is the ring of [[p-adic integers]] and the [[profinite completion]] of '''Z''', respectively
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| :<math>\mathbb Z_p := \varprojlim_n \mathbb Z / p^n</math> and <math>\hat{\mathbb Z} := \varprojlim_n \mathbb Z / n.</math><ref>These two groups play a central role for maximal [[abelian extension]] of [[number field]]s, see [[Kronecker–Weber theorem]]</ref>
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| Most of the facts from finite groups can be generalized directly to the profinite case.<ref>For example the [[Sylow theorems]].</ref>
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| Certain [[Ascending chain condition|conditions on chains of subgroups]], parallel to the notion of [[Noetherian ring|Noetherian]] and [[Artinian ring]]s, allow to deduce further properties. For example the ''[[Krull-Schmidt theorem]]'' states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
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| Another, yet slightly weaker, level of finiteness is the following: a subset ''A'' of ''G'' is said to [[generating set of a group|generate]] the group if any element ''h'' can be written as the product of elements of ''A''. A group is said to be [[finitely generated group|finitely generated]] if it is possible to find a finite subset ''A'' generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.
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| ==Abelian groups==
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| The [[category of groups]] can be subdivided in several ways. A particularly well-understood class of groups are the so-called [[abelian group|abelian]] (in honor of [[Niels Abel]]) or [[commutative]] groups, i.e. the ones satisfying
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| :<math>\forall a,b \in G,\ a*b = b*a \mbox{.}</math>
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| Another way of saying this is that the [[commutator]]
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| :<math>[a,b] =\,\! :\, a^{-1} b^{-1} ab</math>
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| equals the identity element for all ''a'' and ''b''. A ''non-abelian'' group is a group that is not abelian. Even more particular, [[cyclic group|cyclic]] groups are the groups [[Generating set of a group|generated]] by a single element. Being either isomorphic to '''Z''' or to '''Z'''<sub>''n''</sub>, the integers [[modular arithmetic|modulo]] ''n'', they are always abelian. Any [[finitely generated abelian group]] is [[Fundamental theorem of finitely generated abelian groups|known]] to be a [[direct sum of groups]] of these two types. The category of abelian groups is an [[abelian category]]. In fact, abelian groups serve as the prototype of abelian categories. A converse is given by [[Mitchell's embedding theorem]].
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| ==Normal series==
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| Most of the notions developed in group theory are designed to tackle non-abelian groups. There are several notions designed to measure how far a group is from being abelian. The [[commutator subgroup]] (or derived group) is the subgroup generated by commutators [''a'', ''b''], whereas the [[center of a group|center]] is the subgroup of elements that commute with every other group element.
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| Given a group ''G'' and a normal subgroup ''N'' of ''G'', denoted ''N'' ⊲ ''G'', there is an [[exact sequence]]:
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| :1 → ''N'' → ''G'' → ''H'' → 1,
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| where 1 denotes the [[trivial group]] and ''H'' is the quotient ''G''/''N''. This permits the decomposition of ''G'' into two smaller pieces. The other way round, given two groups ''N'' and ''H'', a group ''G'' fitting into an exact sequence as above is called an ''extension'' of ''H'' by ''N''. Given ''H'' and ''N'' there are many different group extensions ''G'', which leads to the [[extension problem]]. There is always at least one extension, called the trivial extension, namely the [[Direct sum of groups|direct sum]] {{nowrap|'G'' {{=}} ''N'' ⊕ ''H''}}, but usually there are more. For example, the [[Klein four-group]] is a non-trivial extension of '''Z'''<sub>2</sub> by '''Z'''<sub>2</sub>. This is a first glimpse of [[homological algebra]] and [[Ext functor|''Ext'' functors]].<ref>{{harvnb|Weibel|1994}} <!--describe Ext^1(Z/2, Z/2)--></ref>
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| Many properties for groups, for example being a [[finite group]] or a [[p-group|''p''-group]] (i.e. the order of every element is a power of ''p'') are stable under extensions and sub- and quotient groups, i.e. if ''N'' and ''H'' have the property, then so does ''G'' and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group ''G'' does not have any (non-trivial) normal subgroups, ''G'' is called [[simple group|simple]]. The name is misleading because a simple group can in fact be very complex. An example is the [[monster group]], whose [[order (group theory)|order]] is about 10<sup>54</sup>. The finite simple groups are known and [[classification of finite simple groups|classified]].
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| [[Induction (mathematics)|Repeatedly]] taking normal subgroups (if they exist) leads to ''[[normal series]]'':
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| :1 = ''G''<sub>0</sub> ⊲ ''G''<sub>1</sub> ⊲ ... ⊲ ''G''<sub>n</sub> = ''G'',
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| i.e. any ''G''<sub>''i''</sub> is a normal subgroup of the next one ''G''<sub>''i''+1</sub>. A group is ''[[solvable group|solvable]]'' (or ''soluble'') if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients ''G''<sub>''i''+1</sub> / ''G''<sub>''i''</sub>, one obtains [[central series]] which lead to ''[[nilpotent group]]s''. They are an approximation of abelian groups in the sense that
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| :<code>[...[[''g''<sub>1</sub>, ''g''<sub>2</sub>], ''g''<sub>3</sub>] ..., ''g''<sub>''n''</sub>]=1</code>
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| for all choices of group elements ''g''<sub>''i''</sub>.
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| There may be distinct normal series for a group ''G''. If it is impossible to refine a given series by inserting further normal subgroups, it is called ''[[composition series]]''. By the ''[[Jordan–Hölder theorem]]'' any two composition series of a given group are equivalent.<ref>This can be shown using the [[Schreier refinement theorem]].</ref>
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| ==Other notions==
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| '''[[General linear group]]''', denoted by GL(''n'', ''F''), is the group of <math>n</math>-by-<math>n</math> [[invertible matrix|invertible matrices]], where the elements of the matrices are taken from a [[field (mathematics)|field]] <math>F</math> such as the real numbers or the complex numbers.
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| '''[[Group representation]]''' (not to be confused with the ''presentation'' of a group). A ''group representation'' is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible [[matrix (mathematics)|matrices]] which is much easier to study.
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| ==Notes==
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| <references/>
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| ==References==
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| *{{cite book | author=Rotman, Joseph | title=An introduction to the theory of groups | location=New York | publisher=Springer-Verlag | year=1994 | isbn=0-387-94285-8}} A standard contemporary reference.
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| * {{Weibel IHA}}
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| * {{cite book | last=Shatz | first=Stephen S. | title=Profinite groups, arithmetic, and geometry | series=Annals of Mathematics Studies | volume=67 | location=Princeton, NJ | publisher=[[Princeton University Press]] | year=1972 | isbn=0-691-08017-8 | zbl=0236.12002 | mr=0347778 | ref=harv }}
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| {{DEFAULTSORT:Glossary Of Group Theory}}
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| [[Category:Group theory| ]]
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| [[Category:Glossaries of mathematics|Group theory]]
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