Editing Glossary of group theory (section)

== 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}}