are precisely the kernels of group homomorphisms with domain
, which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.
is called a normal subgroup of
[3] The usual notation for this relation is
[6] Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup
, hence the fact that all subgroups of an abelian group are normal).
[11] This illustrates the general fact that any subgroup
matrices with real entries under the operation of matrix multiplication and its subgroup
matrices of determinant 1 (the special linear group).
[a] 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.
[13] 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 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.
form a lattice under subset inclusion with least element,
The meet of two normal subgroups,
, in this lattice is their intersection and the join is their product.
The lattice is complete and modular.
is a normal subgroup, we can define a multiplication on cosets as follows:
This relation defines a mapping
To show that this mapping is well-defined, one needs to prove that the choice of representative elements
To this end, consider some other representative elements
This proves that this product is a well-defined mapping between cosets.
With this operation, the set of cosets is itself a group, called the quotient group and denoted with
We call the preimage of the trivial group
the kernel of the homomorphism and denote it by
As it turns out, the kernel is always normal and the image of
[24] In fact, this correspondence is a bijection between the set of all quotient groups of
, and the set of all homomorphic images of
[25] It is also easy to see that the kernel of the quotient map,
itself, so the normal subgroups are precisely the kernels of homomorphisms with domain