Normalsubgrupper

Normalsubgrupper, known in English as normal subgroups, are a central concept in group theory. Given a group G, a subgroup N ≤ G is called normal if g N g^{-1} = N for every g in G. Equivalently, N is invariant under conjugation by all elements of G, or all left cosets gN coincide with the right cosets Ng. Another equivalent formulation is that N is the kernel of a group homomorphism: there exists a homomorphism φ: G → H with N = ker φ.

Quotient groups: If N is normal in G, the set of cosets G/N forms a group with the product (aN)(bN) = (ab)N. The quotient group G/N encodes the structure of G modulo N and is a fundamental construction in many areas of algebra. If N is normal and N ⊆ M ⊆ G, then M/N is a subgroup of G/N; conversely, normal subgroups of G correspond to kernels of homomorphisms from G.

Examples: In any abelian group, every subgroup is normal. In the symmetric group S3, the subgroup A3 is normal of order 3, while the subgroups generated by transpositions have order 2 and are not normal.

Properties: The intersection of any collection of normal subgroups is normal; the product NM of two normal subgroups is normal. The normal subgroups of G form a lattice under inclusion, and the quotient by a normal subgroup is a standard way to study the structure of G. A group with no nontrivial normal subgroups is called simple.

Applications: Normal subgroups are central to the first and higher isomorphism theorems and to the construction of quotient groups, which simplify the analysis of group homomorphisms and group structure.