Évariste Galois was the first to realize the importance of the existence of normal subgroups.
A subgroup ''N'' of a group ''G'' is called a if it is invariant under Conjugation ; that is, for each element ''n'' in ''N'' and each ''g'' in ''G'', the element ''gng''−1 is still in ''N''. We write
:
The following conditions are Equivalent to requiring that a subgroup ''N'' be normal in ''G''. Any one of them may be taken as the definition:
#For all ''g'' in ''G'', ''gNg''−1 ⊆ ''N''.
#For all ''g'' in ''G'', ''gNg''−1 = ''N''.
#The sets of left and right Coset s of ''N'' in ''G'' coincide.
#For all ''g'' in ''G'', ''gN'' = ''Ng''.
#''N'' is a Union of Conjugacy Class es of ''G''.
#There is some Homomorphism on ''G'' for which ''N'' is the Kernel .
Note that condition (1) is logically weaker than condition (2), and condition (3) is logically weaker than condition (4). For this reason, conditions (1) and (3) are often used to prove that ''N'' is normal in ''G'', while conditions (2) and (4) are used to prove consequences of the normality of ''N'' in ''G''.
- {''e''} and ''G'' are always normal subgroups of ''G''. If these are the only ones, then ''G'' is said to be Simple .
- All subgroups ''N'' of an Abelian Group ''G'' are normal, because ''gN'' = ''Ng''. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian Group .
- The Translation Group in any dimension is a normal subgroup of the Euclidean Group ; for example in 3D rotating, translating, and rotating back results in only translation; also reflecting, translating, and reflecting again results in only translation (a translation seen in a mirror looks like a translation, with a reflected translation vector). The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.
- In the Rubik's Cube Group , the subgroup consisting of operations which only affect the corner pieces is normal, because no conjugate transformation can make such an operation affect an edge piece instead of a corner. By contrast, the subgroup consisting of turns of the top face only is not normal, because a conjugate transformation can move parts of the top face to the bottom and hence not all conjugates of elements of this subgroup are contained in the subgroup.
- Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.
- Normality is preserved on taking direct products
- 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 . However, a Characteristic Subgroup of a normal subgroup is normal. Also, a normal subgroup of a Central Factor is normal. In particular, a normal subgroup of a Direct Factor is normal.
- Every subgroup of Index 2 is normal. More generally, a subgroup ''H'' of finite index ''n'' in ''G'' contains a subgroup ''K'' normal in ''G'' and of index dividing ''n''! called the Normal Core . In particular, if ''p'' is the smallest prime dividing the order of ''G'', then every subgroup of index ''p'' is normal.
The normal subgroups of a group ''G'' form a Lattice under Subset Inclusion with Least Element {''e''} and Greatest Element ''G''. Given two normal subgroups ''N'' and ''M'' in ''G'', Meet is defined as
:
and Join is defined as
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