Normal subgroup
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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:
- <math>N\triangleleft G<math>.
There are a number of conditions which are equivalent to requiring that a subgroup N be normal in G. Any one of them may be taken as the definition:
- For each g in G, g−1Ng is contained in N.
- For each g in G, g−1Ng = N.
- The sets of left and right cosets of N in G coincide.
- For each g in G, gN = Ng.
- N is a union of conjugacy classes of G.
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 g−1(Ng) = g−1(gN) = (g−1g)N = 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 normal 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: if N is normal, we can define a multiplication on cosets by
- (a1N)(a2N) := (a1a2)N
This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f : G → G/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset eN = N.
In general, a group homomorphism f: G → H sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e} in H 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/N of G and the set of all homomorphic images of G (up to isomorphism).
See also
de:Normalteiler fr:Sous-groupe distingué it:Sottogruppo normale