Editing Glossary of group theory (section)

== 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}}