Editing Glossary of group theory

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A [[group (mathematics)|group]] is a set together with an [[associative]] operation that admits an [[identity element]] and such that there exists an [[inverse element|inverse]] for every element.

Throughout this glossary, we use {{math|''e''}} to denote the identity element of a group.

{{Compact ToC|name=no|b=|e=|j=|k=|m=|u=|v=|w=|x=|y=|z=|seealso=yes}}

== A ==
{{glossary}}
{{term|1=abelian group}}
{{defn|1=A group {{math|(''G'', •)}} is [[Abelian group|abelian]] if {{math|•}} is commutative, i.e. {{math|1=''g'' • ''h'' = ''h'' • ''g''}} for all {{math|''g'', ''h'' &isin; ''G''}}. Likewise, a group is ''nonabelian'' if this relation fails to hold for any pair {{math|''g'', ''h'' &isin; ''G''}}.}}
{{term|1=ascendant subgroup}}
{{defn|1=A {{gli|subgroup}} {{math|''H''}} of a group {{math|''G''}} is [[ascendant subgroup|ascendant]] if there is an ascending {{gli|subgroup series}} starting from {{math|''H''}} and ending at {{math|''G''}}, such that every term in the series is a {{gli|normal subgroup}} of its successor. The series may be infinite. If the series is finite, then the subgroup is {{gli|subnormal subgroup|subnormal}}.}}
{{term|1=automorphism}}
{{defn|1=An [[group automorphism|automorphism]] of a group is an {{gli|isomorphism}} of the group to itself.}}
{{glossary end}}

== C ==
{{glossary}}
{{term|1=center of a group}}
{{defn|1=The [[center of a group]] {{math|''G''}}, denoted {{math|Z(''G'')}}, is the set of those group elements that commute with all elements of {{math|''G''}}, that is, the set of all {{math|''h'' ∈ ''G''}} such that {{math|1=''hg'' = ''gh''}} for all {{math|''g'' ∈ ''G''}}. {{math|Z(''G'')}} is always a {{gli|normal subgroup}} of {{math|''G''}}. A group&nbsp;{{math|''G''}} is {{gli|Abelian group|abelian}} if and only if {{math|1=Z(''G'') = ''G''}}.}}
{{term|centerless group}}
{{defn|1=A group {{math|''G''}} is [[centerless group|centerless]] if its {{gli|center of a group|center}} {{math|Z(''G'')}} is {{gli|trivial group|trivial}}.}}
{{term|1=central subgroup}}
{{defn|1=A {{gli|subgroup}} of a group is a [[central subgroup]] of that group if it lies inside the {{gli|center of a group|center of the group}}.}}
{{term|centralizer}}
{{defn|1=For a subset {{math|''S''}} of a group&nbsp;{{math|''G''}}, the [[centralizer and normalizer|centralizer]] of {{math|''S''}} in {{math|''G''}}, denoted {{math|C<sub>''G''</sub>(''S'')}}, is the subgroup of {{math|''G''}} defined by
: <math>\mathrm{C}_G(S)=\{ g \in G \mid gs=sg \text{ for all } s \in S\}.</math>}}
{{term|1=characteristic subgroup}}
{{defn|1=A {{gli|subgroup}} of a group is a [[characteristic subgroup]] of that group if it is mapped to itself by every {{gli|automorphism}} of the parent group.}}
{{term|1=characteristically simple group}}
{{defn|1=A group is said to be [[Characteristically simple group|characteristically simple]] if it has no proper nontrivial {{gli|characteristic subgroup|characteristic subgroups}}.}}
{{term|class function}}
{{defn|1=A [[class function]] on a group {{math|''G''}} is a function that it is constant on the {{gli|conjugacy class|conjugacy classes}} of {{math|''G''}}.}}
{{term|class number}}
{{defn|1=The [[class number (group theory)|class number]] of a group is the number of its {{gli|conjugacy class|conjugacy classes}}.}}
{{term|commutator}}
{{defn|1=The [[commutator (group theory)|commutator]] of two elements {{math|''g''}} and {{math|''h''}} of a group&nbsp;{{math|''G''}} is the element {{math|1=[''g'', ''h''] = ''g''<sup>−1</sup>''h''<sup>−1</sup>''gh''}}. Some authors define the commutator as {{math|1=[''g'', ''h''] = ''ghg''<sup>−1</sup>''h''<sup>−1</sup>}} instead. The commutator of two elements {{math|''g''}} and {{math|''h''}} is equal to the group's identity if and only if {{math|''g''}} and {{math|''h''}} commutate, that is, if and only if {{math|1=''gh'' = ''hg''}}.}}
{{term|commutator subgroup}}
{{defn|1=The [[commutator subgroup]] or derived subgroup of a group is the subgroup [[Generating set of a group|generated]] by all the {{gli|commutator|commutators}} of the group.}}
{{term|1=complete group}}
{{defn|1=A group {{math|''G''}} is said to be [[Complete group|complete]] if it is {{gli|centerless group|centerless}} and if every {{gli|automorphism}} of {{math|''G''}} is an [[inner automorphism]].}}
{{term|composition series}}
{{defn|1=A [[composition series (group theory)|composition series]] of a group {{math|''G''}} is a [[subnormal series]] of finite length
: <math>1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,</math>
with strict inclusions, such that each {{math|''H''<sub>''i''</sub>}} is a maximal strict {{gli|normal subgroup}} of {{math|''H''<sub>''i''+1</sub>}}. Equivalently, a composition series is a subnormal series such that each {{gli|factor group}} {{math|''H''<sub>''i''+1</sub> / ''H''<sub>''i''</sub>}} is {{gli|simple group|simple}}. The factor groups are called composition factors.}}
{{term|1=conjugacy-closed subgroup}}
{{defn|1=A {{gli|subgroup}} of a group is said to be [[Conjugacy-closed subgroup|conjugacy-closed]] if any two elements of the subgroup that are {{gli|conjugate elements|conjugate}} in the group are also conjugate in the subgroup.}}
{{term|conjugacy class}}
{{defn|1=The [[conjugacy classes]] of a group {{math|''G''}} are those subsets of {{math|''G''}} containing group elements that are {{gli|conjugate elements|conjugate}} with each other.}}
{{term|conjugate elements}}
{{defn|1=Two elements {{math|''x''}} and {{math|''y''}} of a group&nbsp;{{math|''G''}} are [[conjugate (group theory)|conjugate]] if there exists an element {{math|''g'' ∈ ''G''}} such that {{math|1=''g''<sup>−1</sup>''xg'' = ''y''}}. The element {{math|''g''<sup>−1</sup>''xg''}}, denoted {{math|''x''<sup>''g''</sup>}}, is called the conjugate of {{math|''x''}} by {{math|''g''}}. Some authors define the conjugate of {{math|''x''}} by {{math|''g''}} as {{math|''gxg''<sup>−1</sup>}}. This is often denoted {{math|<sup>''g''</sup>''x''}}. Conjugacy is an [[equivalence relation]]. Its [[equivalence class]]es are called [[conjugacy class]]es.}}
{{term|1=conjugate subgroups}}
{{defn|1=Two subgroups {{math|''H''<sub>1</sub>}} and {{math|''H''<sub>2</sub>}} of a group {{math|''G''}} are [[conjugate subgroups]] if there is a {{math|''g'' ∈ ''G''}} such that {{math|1=''gH''<sub>1</sub>''g''<sup>−1</sup> = ''H''<sub>2</sub>}}.}}
{{term|1=contranormal subgroup}}
{{defn|1=A {{gli|subgroup}} of a group {{math|''G''}} is a [[contranormal subgroup]] of {{math|''G''}} if its {{gli|normal closure}} is {{math|''G''}} itself.}}
{{term|1=cyclic group}}
{{defn|1=A [[cyclic group]] is a group that is [[Generating set of a group|generated]] by a single element, that is, a group such that there is an element {{math|''g''}} in the group such that every other element of the group may be obtained by repeatedly applying the group operation to&nbsp;{{math|''g''}} or its inverse.}}
{{glossary end}}

== D ==
{{glossary}}
{{term|1=derived subgroup}}
{{defn|1=Synonym for {{gli|commutator subgroup}}.}}
{{term|direct product}}
{{defn|1=The [[Direct product of groups|direct product]] of two groups {{math|''G''}} and {{math|''H''}}, denoted {{math|''G'' × ''H''}}, is the [[cartesian product]] of the underlying sets of {{math|''G''}} and {{math|''H''}}, equipped with a component-wise defined binary operation {{math|1=(''g''<sub>1</sub>, ''h''<sub>1</sub>) · (''g''<sub>2</sub>, ''h''<sub>2</sub>) = (''g''<sub>1</sub> ⋅ ''g''<sub>2</sub>, ''h''<sub>1</sub> ⋅ ''h''<sub>2</sub>)}}. With this operation, {{math|''G'' × ''H''}} itself forms a group.}} 
{{glossary end}}

== E ==
{{glossary}}
{{term|exponent of a group}}
{{defn|The exponent of a group {{math|''G''}} is the smallest positive integer {{math|''n''}} such that {{math|1=''g''<sup>''n''</sup> = ''e''}} for all {{math|''g'' ∈ ''G''}}. It is the [[least common multiple]] of the {{gli|order of a group|orders}} of all elements in the group. If no such positive integer exists, the exponent of the group is said to be infinite.}}
{{glossary end}}

== F ==
{{glossary}}
{{term|factor group}}
{{defn|Synonym for {{gli|quotient group}}.}}
{{term|1=FC-group}}
{{defn|1=A group is an [[FC-group]] if every {{gli|conjugacy class}} of its elements has finite cardinality.}}
{{term|finite group}}
{{defn|A [[finite group]] is a group of finite {{gli|order of a group|order}}, that is, a group with a finite number of elements.}}
{{term|1=finitely generated group}}
{{defn|1=A group {{math|''G''}} is [[finitely generated group|finitely generated]] if there is a finite {{gli|generating set}}, that is, if there is a finite set {{math|''S''}} of elements of {{mvar|G}} such that every element of {{math|''G''}} can be written as the combination of finitely many elements of {{math|''S''}} and of inverses of elements of {{math|''S''}}.}}
{{glossary end}}

== G == 
{{glossary}}
{{term|1=generating set}}
{{defn|A [[generating set of a group|generating set]] of a group {{math|''G''}} is a subset {{math|''S''}} of {{math|''G''}} such that every element of {{math|''G''}} can be expressed as a combination (under the group operation) of finitely many elements of {{math|''S''}} and inverses of elements of {{math|S}}. Given a subset {{math|''S''}} of {{math|''G''}}.  We denote by {{math|{{angbr|''S''}}}} the smallest subgroup of {{math|''G''}} containing {{math|''S''}}. {{math|{{angbr|''S''}}}} is called the subgroup of {{math|''G''}} generated by {{math|''S''}}.}}
{{term|1=group automorphism}}
{{defn|See {{gli|automorphism}}.}}
{{term|1=group homomorphism}}
{{defn|See {{gli|homomorphism}}.}}
{{term|1=group isomorphism}}
{{defn|See {{gli|isomorphism}}.}}
{{glossary end}}

== H ==
{{glossary}}
{{term|1=homomorphism}}
{{defn|1=Given two groups {{math|(''G'', •)}} and {{math|(''H'', ·)}}, a [[group homomorphism|homomorphism]] from {{math|''G''}} to {{math|''H''}} is a [[function (mathematics)|function]] {{math|''h'' : ''G'' → ''H''}} such that for all {{math|''a''}} and {{math|''b''}} in {{math|''G''}}, {{math|1=''h''(''a'' • ''b'') = ''h''(''a'') · ''h''(''b'')}}.}}
{{glossary end}}

== I ==
{{glossary}}
{{term|1=index of a subgroup}}
{{defn|1=The [[index of a subgroup|index]] of a {{gli|subgroup}} {{math|''H''}} of a group {{math|''G''}}, denoted {{math|{{abs|''G'' : ''H''}}}} or {{math|{{bracket|''G'' : ''H''}}}} or {{math|(''G'' : ''H'')}}, is the number of [[coset]]s of {{math|''H''}} in {{math|''G''}}. For a {{gli|normal subgroup}} {{math|''N''}} of a group {{math|''G''}}, the index of {{math|''N''}} in {{math|''G''}} is equal to the {{gli|order of a group|order}} of the {{gli|quotient group}} {{math|''G'' / ''N''}}. For a {{gli|finite group|finite}} subgroup {{math|''H''}} of a finite group {{math|''G''}}, the index of {{math|''H''}} in {{math|''G''}} is equal to the quotient of the orders of {{math|''G''}} and {{math|''H''}}.}}
{{term|1=isomorphism}}
{{defn|1=Given two groups {{math|(''G'', •)}} and {{math|(''H'', ·)}}, an [[group isomorphism|isomorphism]] between {{math|''G''}} and {{math|''H''}} is a [[bijection|bijective]] {{gli|homomorphism}} from {{math|''G''}} to {{math|''H''}}, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups are ''isomorphic'' if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.}}
{{glossary end}}

== L ==
{{glossary}}
{{term|1=lattice of subgroups}}
{{defn|1=The [[lattice of subgroups]] of a group is the [[Lattice (order)|lattice]] defined by its {{gli|subgroup|subgroups}}, [[Partially ordered set|partially ordered]] by [[set inclusion]].}}
{{term|1=locally cyclic group}}
{{defn|1=A group is [[locally cyclic group|locally cyclic]] if every {{gli|finitely generated group|finitely generated}} subgroup is {{gli|cyclic group|cyclic}}. Every cyclic group is locally cyclic, and every {{gli|finitely generated group|finitely-generated}} locally cyclic group is cyclic. Every locally cyclic group is {{gli|abelian group|abelian}}. Every {{gli|subgroup}}, every {{gli|quotient group}} and every {{gli|group homomorphism|homomorphic}} image of a locally cyclic group is locally cyclic.}}
{{glossary end}}

== N ==
{{glossary}}
{{term|1=no small subgroup}}
{{defn|1=A [[topological group]] has [[no small subgroup]] if there exists a neighborhood of the identity element that does not contain any nontrivial subgroup.}}
{{term|normal closure}}
{{defn|1=The [[normal closure (group theory)|normal closure]] of a subset&nbsp;{{math|''S''}} of a group&nbsp;{{math|''G''}} is the intersection of all {{gli|normal subgroup|normal subgroups}} of&nbsp;{{math|''G''}} that contain&nbsp;{{math|''S''}}.}}
{{term|1=normal core}}
{{defn|1=The [[normal core]] of a {{gli|subgroup}} {{math|''H''}} of a group {{math|''G''}} is the largest {{gli|normal subgroup}} of {{math|''G''}} that is contained in {{math|''H''}}.}}
{{term|normal series}}
{{defn|1=A [[normal series]] of a group&nbsp;{{math|''G''}} is a sequence of {{gli|normal subgroup|normal subgroups}} of {{math|''G''}} such that each element of the sequence is a normal subgroup of the next element:
: <math>1 = A_0\triangleleft A_1\triangleleft \cdots \triangleleft A_n = G</math>
with
: <math>A_i\triangleleft G</math>.}}
{{term|1=normal subgroup}}
{{defn|1=A {{gli|subgroup}} {{math|''N''}} of a group {{math|''G''}} is [[Normal subgroup|normal]] in {{math|''G''}} (denoted {{math|''N'' ◅ ''G''}}) if the {{gli|conjugate elements|conjugation}} of an element {{math|''n''}} of {{math|''N''}} by an element {{math|''g''}} of {{math|''G''}} is always in {{math|''N''}}, that is, if for all {{math|''g'' ∈ ''G''}} and {{math|''n'' ∈ ''N''}}, {{math|''gng''{{sup|−1}} ∈ ''N''}}. A normal subgroup {{math|''N''}} of a group {{math|''G''}} can be used to construct the {{gli|quotient group}} {{math|''G'' / ''N''}}.}}
{{term|normalizer}}
{{defn|1=For a subset {{math|''S''}} of a group&nbsp;{{math|''G''}}, the [[centralizer and normalizer|normalizer]] of {{math|''S''}} in {{math|''G''}}, denoted {{math|N<sub>''G''</sub>(''S'')}}, is the subgroup of {{math|''G''}} defined by
: <math>\mathrm{N}_G(S)=\{ g \in G \mid gS=Sg \}.</math>}}
{{glossary end}}

== O ==
{{glossary}}
{{term|1=orbit}}
{{defn|1=Consider a group {{math|''G''}} acting on a set {{math|''X''}}. The [[Orbit (group theory)|orbit]] of an element {{math|''x''}} in {{math|''X''}} is the set of elements in {{math|''X''}} to which {{math|''x''}} can be moved by the elements of {{math|''G''}}. The orbit of {{math|''x''}} is denoted by {{math|''G'' ⋅ ''x''}}}}
{{term|1=order of a group}}
{{defn|1=The [[order of a group]] {{math|(''G'', •)}} is the [[Cardinal number|cardinality]] (i.e. number of elements) of {{math|G}}.  A group with finite order is called a [[finite group]].}}
{{term|1=order of a group element}}
{{defn|1=The [[Order of a group element|order of an element]] {{math|''g''}} of a group {{math|''G''}} is the smallest [[positive number|positive]] [[integer]] {{math|''n''}} such that {{math|1=''g''<sup>''n''</sup> = ''e''}}. If no such integer exists, then the order of {{math|''g''}} is said to be infinite. The order of a finite group is [[divisible]] by the order of every element.}}
{{glossary end}}

== P ==
{{glossary}}
{{term|perfect core}}
{{defn|1=The [[perfect core]] of a group is its largest {{gli|perfect group|perfect}} subgroup.}}
{{term|perfect group}}
{{defn|1=A [[perfect group]] is a group that is equal to its own {{gli|commutator subgroup}}.}}
{{term|1=periodic group}}
{{defn|1=A group is [[periodic group|periodic]] if every group element has finite {{gli|order of a group element|order}}. Every [[finite group]] is periodic.}}
{{term|1=permutation group}}
{{defn|1=A [[permutation group]] is a group whose elements are [[permutation]]s of a given [[Set (mathematics)|set]] {{math|''M''}} (the [[bijective function]]s from set {{math|''M''}} to itself) and whose [[group operation]] is the [[function composition|composition]] of those permutations. The group consisting of all permutations of a set {{math|''M''}} is the [[symmetric group]] of {{math|''M''}}.}}
{{term|1=''p''-group}}
{{defn|1=If {{math|''p''}} is a [[prime number]], then a [[p-group|{{math|''p''}}-group]] is one in which the order of every element is a power of {{math|''p''}}. A finite group is a {{math|''p''}}-group if and only if the {{gli|order of a group|order}} of the group is a power of {{math|''p''}}.}}
{{term|1=''p''-subgroup}}
{{defn|1=A {{gli|subgroup}} that is also a {{gli|''p''-group|{{math|''p''}}-group}}. The study of {{math|''p''}}-subgroups is the central object of the [[Sylow theorems]].}}
{{glossary end}}

== Q ==
{{glossary}}
{{term|1=quotient group}}
{{defn|1=Given a group {{math|''G''}} and a {{gli|normal subgroup}} {{math|''N''}} of {{math|''G''}}, the [[quotient group]] is the set {{math|''G'' / ''N''}} of [[left coset]]s {{math|{{mset|''aN'' : ''a'' &isin; ''G''}}}} together with the operation {{math|1=''aN'' • ''bN'' = ''abN''}}. The relationship between normal subgroups, homomorphisms, and factor groups is summed up in the [[fundamental theorem on homomorphisms]].}}
{{glossary end}}

== R ==
{{glossary}}
{{term|real element}}
{{defn|1=An element {{math|''g''}} of a group {{math|''G''}} is called a [[real element]] of {{math|''G''}} if it belongs to the same {{gli|conjugacy class}} as its inverse, that is, if there is a {{math|''h''}} in {{math|''G''}} with&nbsp;{{math|1=''g''<sup>''h''</sup> = ''g''<sup>−1</sup>}}, where {{math|''g''<sup>''h''</sup>}} is defined as {{math|''h''<sup>−1</sup>''gh''}}. An element of a group {{math|''G''}} is real if and only if for all [[group representation|representations]] of {{math|''G''}} the [[trace (linear algebra)|trace]] of the corresponding matrix is a real number.}}
{{glossary end}}

== S ==
{{glossary}}
{{term|1=serial subgroup}}
{{defn|1=A {{gli|subgroup}} {{math|''H''}} of a group {{math|''G''}} is a [[serial subgroup]] of {{math|''G''}} if there is a chain {{math|''C''}} of subgroups of {{math|''G''}} from {{math|''H''}} to {{math|''G''}} such that for each pair of consecutive subgroups {{math|''X''}} and {{math|''Y''}} in {{math|''C''}}, {{math|''X''}} is a {{gli|normal subgroup}} of {{math|''Y''}}. If the chain is finite, then {{math|''H''}} is a {{gli|subnormal subgroup}} of {{math|''G''}}.}}
{{term|1=simple group}}
{{defn|1=A [[simple group]] is a {{gli|trivial group|nontrivial group}} whose only {{gli|normal subgroup|normal subgroups}} are the trivial group and the group itself.}}
{{term|1=subgroup}}
{{defn|1=A [[subgroup]] of a group {{math|''G''}} is a [[subset]] {{math|''H''}} of the elements of {{math|''G''}} that itself forms a group when equipped with the restriction of the [[group operation]] of {{math|''G''}} to {{math|''H'' × ''H''}}. A subset {{math|''H''}} of a group {{math|''G''}} is a subgroup of {{math|''G''}} if and only if it is nonempty and [[Closure (mathematics)|closed]] under products and inverses, that is, if and only if for every {{math|''a''}} and {{math|''b''}} in {{math|''H''}}, {{math|''ab''}} and {{math|''a''<sup>−1</sup>}} are also in {{math|''H''}}.}}
{{term|subgroup series}}
{{defn|1=A [[subgroup series]] of a group {{math|''G''}} is a sequence of {{gli|subgroup|subgroups}} of {{math|''G''}} such that each element in the series is a subgroup of the next element:
: <math>1 = A_0 \leq A_1 \leq \cdots \leq A_n = G.</math>}}
{{term|1=subnormal subgroup}}
{{defn|1=A {{gli|subgroup}} {{math|''H''}} of a group {{math|''G''}} is a [[subnormal subgroup]] of {{math|''G''}} if there is a finite chain of subgroups of the group, each one {{gli|normal subgroup|normal}} in the next, beginning at {{math|''H''}} and ending at {{math|''G''}}. }}
{{term|1=symmetric group}}
{{defn|1=Given a set {{math|''M''}}, the [[symmetric group]] of {{math|''M''}} is the set of all [[permutation]]s of {{math|''M''}} (the set all [[bijective function]]s from {{math|''M''}} to {{math|''M''}}) with the [[function composition|composition]] of the permutations as group operation. The symmetric group of a [[finite set]] of size {{math|''n''}} is denoted {{math|S<sub>''n''</sub>}}. (The symmetric groups of any two sets of the same size are [[group isomorphism|isomorphic]].)}}
{{glossary end}}

== T ==
{{glossary}}
{{term|1=torsion group}}
{{defn|1=Synonym for {{gli|periodic group}}.}}
{{term|1=transitively normal subgroup}}
{{defn|1=A {{gli|subgroup}} of a group is said to be [[Transitively normal subgroup|transitively normal]] in the group if every {{gli|normal subgroup}} of the subgroup is also normal in the whole group.}}
{{term|1=trivial group}}
{{defn|1=A [[trivial group]] is a group consisting of a single element, namely the identity element of the group. All such groups are [[group isomorphism|isomorphic]], and one often speaks of ''the'' trivial group.}}
{{glossary end}}

== Basic definitions ==
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]].

'''[[kernel (algebra)|Kernel]] of a group homomorphism'''. It is the [[preimage]] of the identity in the [[codomain]] of a group homomorphism.  Every normal subgroup is the kernel of a group homomorphism and vice versa.

'''[[Direct product]]''', '''[[direct sum of groups|direct sum]]''', and '''[[semidirect product]]''' of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation.

== Types of groups ==
'''[[Finitely generated group]]'''. If there exists a finite set {{math|''S''}} such that {{math|1={{angbr|''S''}} = ''G''}}, then {{math|''G''}} is said to be [[generating set of a group|finitely generated]]. If {{math|''S''}} can be taken to have just one element, {{math|''G''}} is a [[cyclic group]] of finite order, an [[infinite cyclic group]], or possibly a group {{math|{{mset|''e''}}}} with just one element.

'''[[Simple group]]'''. Simple groups are those groups having only {{math|''e''}} and themselves as [[normal subgroup]]s. 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>. Every finite group is built up from simple groups via [[extension problem|group extensions]], so the study of finite simple groups is central to the study of all finite groups. The finite simple groups are known and [[classification of finite simple groups|classified]].

The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of [[cyclic group|cyclic]] p-groups.
This can be extended to a complete classification of all [[finitely generated abelian group]]s, that is all abelian groups that are [[generating set of a group|generated]] by a finite set.

The situation is much more complicated for the non-abelian groups.

'''[[Free group]]'''. Given any set {{math|''A''}}, one can define a group as the smallest group containing the [[free semigroup]] of {{math|''A''}}. The group consists of the finite strings (words) that can be composed by elements from {{math|''A''}}, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance {{math|1=(''abb'') • (''bca'') = ''abbbca''}}.

Every group {{math|(''G'', •)}} is basically a factor group of a free group generated by {{math|''G''}}.  Refer to ''[[Presentation of a group]]'' for more explanation.
One can then ask [[algorithm]]ic questions about these presentations, such as:
* Do these two presentations specify isomorphic groups?; or
* Does this presentation specify the trivial group?
The general case of this is the [[word problem for groups|word problem]], and several of these questions are in fact unsolvable by any general algorithm.

'''[[General linear group]]''', denoted by {{math|GL(''n'', ''F'')}}, is the group of {{math|''n''}}-by-{{math|''n''}} [[invertible matrix|invertible matrices]], where the elements of the matrices are taken from a [[field (mathematics)|field]] {{math|''F''}} such as the real numbers or the complex numbers.

'''[[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.

== See also ==
* [[Glossary of Lie groups and Lie algebras]]
* [[Glossary of ring theory]]
* [[Composition series]]
* [[Normal series]]

[[Category:Group theory| ]]
[[Category:Glossaries of mathematics|Group theory]]
[[Category:Wikipedia glossaries using description lists]]