Normal subgroup

In abstract algebra, a normal subgroup is a special kind of subgroup that follows a specific rule. It is a subgroup that stays the same even when you combine its elements with any element from the larger group in a certain way. This special property makes normal subgroups very useful in mathematics.

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

The idea of normal subgroups was first discovered by Évariste Galois, a mathematician who showed how important they are in studying groups. Because of this, normal subgroups play a key role in many areas 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 "flip" 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 commute), 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 transformations 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 (intersection) and their combination (product) are also normal subgroups. These normal subgroups can be organized in a special structure called a lattice, where we can find the smallest and largest normal subgroups, as well as how they combine together. This structure helps mathematicians study groups in a more organized way.

Normal subgroups, quotient groups and homomorphisms

When a subgroup is normal, we can create a new group from it by looking at "cosets." These are groups formed by combining the subgroup with elements of the larger group. This lets us define a new way to multiply these cosets, creating what is called a quotient group.

There is a special mapping, called a homomorphism, between groups that helps us understand how they relate. This mapping sends subgroups of one group to subgroups of another. Importantly, the "kernel" of this mapping—which is like the preimage of the smallest possible subgroup—is always a normal subgroup. This connection shows how normal subgroups are linked to these mappings between groups.