Normal subgroup

In abstract algebra, a normal subgroup is a special type of subgroup. It is a subgroup that looks the same even when you mix its elements with any element from the bigger group in a certain way. This special feature makes normal subgroups very useful in math.

Normal subgroups are important because they help us make new groups called quotient groups from the original group. They are also closely linked to something called the kernels of group homomorphisms. This means they can help us understand and sort out these homomorphisms better.

The idea of normal subgroups was first found by Évariste Galois, a mathematician who showed how useful they are in studying groups. Because of this, normal subgroups are very important in many parts of algebra and group theory.

Definitions

A subgroup is a smaller group inside a bigger group. We call this smaller group a normal subgroup if it stays the same when we mix its elements with elements from the bigger group.

This means that if you take any element from the normal subgroup and change it using an element from the bigger group, you still end up with an element from the normal subgroup. We write this special relationship as ( N \triangleleft G ).

Examples

In group theory, a normal subgroup is a special kind of subgroup. For any group, the smallest subgroup (just the identity element) and the whole group itself are always normal.

When a group is "abelian" (where all elements work together), every subgroup is normal. For example, in the Rubik's Cube group, certain subgroups that only change the corner or edge pieces are normal. Another example is the translation group, which is normal in the Euclidean group, meaning certain moves can be rearranged without changing the result.

Properties

If a smaller group, called a subgroup, is part of a bigger group and this smaller group stays the same no matter how you mix up the bigger group, we call it a normal subgroup. This idea helps us understand how groups can be built from smaller parts.

Normal subgroups follow special rules. For example, if you have two normal subgroups inside a bigger group, their overlap and their combination are also normal subgroups. These normal subgroups can be organized in a special structure. This structure helps mathematicians study groups in a more organized way.

Normal subgroups, quotient groups and homomorphisms

When a subgroup is normal, we can make a new group from it by using "cosets." Cosets are groups made by mixing the subgroup with parts of the larger group. This helps us create a new way to multiply these cosets, forming what is called a quotient group.

There is a special mapping between groups, called a homomorphism, that shows us how groups are related. This mapping links subgroups of one group to subgroups of another. The "kernel" of this mapping—which is like the reverse image of the smallest subgroup—is always a normal subgroup. This shows how normal subgroups connect to these mappings between groups.