normaaliryhmän

Normaaliryhmä, also known as a normal subgroup in English, is a fundamental concept in group theory. A subgroup H of a group G is called a normal subgroup if it is invariant under conjugation by any element of G. This means that for every element g in G and every element h in H, the element g * h * g^(-1) is also in H. The notation for a normal subgroup is H ◁ G.

Normal subgroups are crucial because they allow for the construction of quotient groups. If H is a normal subgroup of G, then the set of left (or right) cosets of H in G forms a group under the operation of coset multiplication. This new group is called the quotient group or factor group of G by H, denoted as G/H. The fact that H is normal ensures that the coset multiplication is well-defined.

Every group G has at least two normal subgroups: the trivial subgroup containing only the identity element, and the group G itself. Groups that have no other normal subgroups are called simple groups. The concept of normal subgroups is essential for understanding the structure of groups and for classifying them. They play a key role in homomorphisms, as the kernel of any group homomorphism is always a normal subgroup.