Editing Normal subgroup (section)

== Properties ==

* If <math>H</math> is a normal subgroup of <math>G</math>, and <math>K</math> is a subgroup of <math>G</math> containing <math>H</math>, then <math>H</math> is a normal subgroup of <math>K</math>.{{sfn|Hungerford|2003|p=42}}
* A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a [[transitive relation]]. The smallest group exhibiting this phenomenon is the [[dihedral group]] of order 8.{{sfn|Robinson|1996|p=17}} However, a [[characteristic subgroup]] of a normal subgroup is normal.{{sfn|Robinson|1996|p=28}} A group in which normality is transitive is called a [[T-group (mathematics)|T-group]].{{sfn|Robinson|1996|p=402}}
* The two groups <math>G</math> and <math>H</math> are normal subgroups of their [[Direct product of groups|direct product]] <math>G \times H</math>.
* If the group <math>G</math> is a [[semidirect product]] <math>G = N \rtimes H</math>, then <math>N</math> is normal in <math>G</math>, though <math>H</math> need not be normal in <math>G</math>.
* If <math>M</math> and <math>N</math> are normal subgroups of an additive group <math>G</math> such that <math>G = M + N</math> and <math>M \cap N = \{0\}</math>, then <math>G = M \oplus N</math>.{{sfn|Hungerford|2013|p=290}}
* Normality is preserved under surjective homomorphisms;{{sfn|Hall|1999|p=29}} that is, if <math>G \to H</math> is a surjective group homomorphism and <math>N</math> is normal in <math>G</math>, then the image <math>f(N)</math> is normal in <math>H</math>.
* Normality is preserved by taking [[Inverse image|inverse images]];{{sfn|Hall|1999|p=29}} that is, if <math>G \to H</math> is a group homomorphism and <math>N</math> is normal in <math>H</math>, then the inverse image <math>f^{-1}(N)</math> is normal in <math>G</math>.
* Normality is preserved on taking [[direct product of groups|direct products]];{{sfn|Hungerford|2003|p=46}} that is, if <math>N_1 \triangleleft G_1</math> and <math>N_2 \triangleleft G_2</math>, then <math>N_1 \times N_2\; \triangleleft \;G_1 \times G_2</math>.
* Every subgroup of [[Index (group theory)|index]] 2 is normal. More generally, a subgroup, <math>H</math>, of finite index, <math>n</math>, in <math>G</math> contains a subgroup, <math>K,</math> normal in <math>G</math> and of index dividing <math>n!</math> called the [[normal core]]. In particular, if <math>p</math> is the smallest prime dividing the order of <math>G</math>, then every subgroup of index <math>p</math> is normal.{{sfn|Robinson|1996|p=36}}
* The fact that normal subgroups of <math>G</math> are precisely the kernels of group homomorphisms defined on <math>G</math> accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is [[Simple group|simple]] if and only if it is isomorphic to all of its non-identity homomorphic images,{{sfn|Dõmõsi|Nehaniv|2004|p=7}} a finite group is [[Perfect group|perfect]] if and only if it has no normal subgroups of prime [[Index of a subgroup|index]], and a group is [[Imperfect group|imperfect]] if and only if the [[derived subgroup]] is not supplemented by any proper normal subgroup.

=== Lattice of normal subgroups ===
Given two normal subgroups, <math>N</math> and <math>M</math>, of <math>G</math>, their intersection <math>N\cap M</math> and their [[Product of subgroups|product]] <math>N M = \{n m : n \in N\; \text{ and }\; m \in M \}</math> are also normal subgroups of <math>G</math>. 

The normal subgroups of <math>G</math> form a [[Lattice (order)|lattice]] under [[subset inclusion]] with [[least element]], <math>\{ e \}</math>, and [[greatest element]], <math>G</math>. The [[Meet (lattice theory)|meet]] of two normal subgroups, <math>N</math> and <math>M</math>, in this lattice is their intersection and the [[Join (lattice theory)|join]] is their product.

The lattice is [[Complete lattice|complete]] and [[Modular lattice|modular]].{{sfn|Hungerford|2003|p=46}}