In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written:
.
Another way to put this is saying that right and left cosets of N in G coincide:
- N g = g g−1 N g = g N for all g in G.
A normal subgroup can also be defined by: A subgroup N of a group G is a normal subgroup if N is a union of conjugacy classes of 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 g−1Ng = g−1gN = N.
The normal subgroups of any group G form a lattice under inclusion. The minimum and maximum elements are {e} and G, the greatest lower bound of two subgroups is their intersection and their least upper bound is a product group .
Normal groups and homomorphisms
Normal subgroups are of relevance because if N is normal, then the factor group G/N may be formed.
Normal subgroups of G are precisely the kernels of group homomorphisms f : G → H.
If H is normal, we can define a multiplication on cosets by
- (a1H)(a2H) := (a1a2)H
This turns the set of cosets into a group called the quotient group G/H. There is a natural homomorphism f : G → G/H given by f(a)=aH. The image f(H) consists only of the identity element of G/H, the coset eH = H.
In general, a group homomorphism f: G → K sends subgroups of G to subgroups of K. Also, the preimage of any subgroup of K is a subgroup of G. We call the preimage of the trivial group {e} in K the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f). In fact, this correspondence is a bijection between the set of all quotient groups G/H and the set of all homomorphic images of G (up to isomorphism).
See also