Editing Normal subgroup (section)

== Examples ==
For any group <math>G</math>, the trivial subgroup <math>\{ e \}</math> consisting of only the identity element of <math>G</math> is always a normal subgroup of <math>G</math>. Likewise, <math>G</math> itself is always a normal subgroup of <math>G</math> (if these are the only normal subgroups, then <math>G</math> is said to be [[Simple group|simple]]).{{sfn|Robinson|1996|p=16}} Other named normal subgroups of an arbitrary group include the [[Center (group theory)|center of the group]] (the set of elements that commute with all other elements) and the [[commutator subgroup]] <math>[G,G]</math>.{{sfn|Hungerford|2003|p=45}}{{sfn|Hall|1999|p=138}} More generally, since conjugation is an isomorphism, any [[characteristic subgroup]] is a normal subgroup.{{sfn|Hall|1999|p=32}}

If <math>G</math> is an [[abelian group]] then every subgroup <math>N</math> of <math>G</math> is normal, because <math>gN = \{gn\}_{n\in N} = \{ng\}_{n\in N} = Ng</math>. More generally, for any group <math>G</math>, every subgroup of the ''center'' <math>Z(G)</math> of <math>G</math> is normal in <math>G</math> (in the special case that <math>G</math> is abelian, the center is all of <math>G</math>, hence the fact that all subgroups of an abelian group are normal). A group that is not abelian but for which every subgroup is normal is called a [[Hamiltonian group]].{{sfn|Hall|1999|p=190}}

A concrete example of a normal subgroup is the subgroup <math>N = \{(1), (123), (132)\}</math> of the [[symmetric group]] <math>S_3</math>, consisting of the identity and both three-cycles. In particular, one can check that every coset of <math>N</math> is either equal to <math>N</math> itself or is equal to <math>(12)N = \{ (12), (23), (13)\}</math>. On the other hand, the subgroup <math>H = \{(1), (12)\}</math> is not normal in <math>S_3</math> since <math>(123)H = \{(123), (13) \} \neq \{(123), (23) \} = H(123)</math>.{{sfn|Judson|2020|loc = Section 10.1}} This illustrates the general fact that any subgroup <math>H \leq G</math> of index two is normal.

As an example of a normal subgroup within a [[matrix group]], consider the [[general linear group]] <math>\mathrm{GL}_n(\mathbf{R})</math> of all invertible <math>n\times n</math> matrices with real entries under the operation of matrix multiplication and its subgroup <math>\mathrm{SL}_n(\mathbf{R})</math> of all <math>n\times n</math> matrices of [[determinant (mathematics)|determinant]] 1 (the [[special linear group]]). To see why the subgroup <math>\mathrm{SL}_n(\mathbf{R})</math> is normal in <math>\mathrm{GL}_n(\mathbf{R})</math>, consider any matrix <math>X</math> in <math>\mathrm{SL}_n(\mathbf{R})</math> and any invertible matrix <math>A</math>. Then using the two important identities <math>\det(AB)=\det(A)\det(B)</math> and <math>\det(A^{-1})=\det(A)^{-1}</math>, one has that <math>\det(AXA^{-1}) = \det(A) \det(X) \det(A)^{-1} = \det(X) = 1</math>, and so <math>AXA^{-1} \in \mathrm{SL}_n(\mathbf{R})</math> as well. This means <math>\mathrm{SL}_n(\mathbf{R})</math> is closed under conjugation in <math>\mathrm{GL}_n(\mathbf{R})</math>, so it is a normal subgroup.{{efn|In other language: <math>\det</math> is a homomorphism from <math>\mathrm{GL}_n(\mathbf{R})</math> to the multiplicative subgroup <math> \mathbf{R}^\times</math>, and <math>\mathrm{SL}_n(\mathbf{R})</math> is the kernel.  Both arguments also work over the [[complex number]]s, or indeed over an arbitrary [[field (mathematics)|field]].}}

In the [[Rubik's Cube group]], the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.{{sfn|Bergvall|Hynning|Hedberg|Mickelin|2010|p=96}}

The [[translation group]] is a normal subgroup of the [[Euclidean group]] in any dimension.{{sfn|Thurston|1997|p=218}} This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all [[Rotation|rotations]] about the origin is ''not'' a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.