A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. In other words, if an element of the larger group is conjugated by an element of the normal subgroup, it remains in the normal subgroup.
Normal subgroups have many important properties and applications in group theory and other areas of mathematics. They provide a way to break down a larger group into smaller, more manageable pieces and can help to understand the structure of a group.
One way to determine if a subgroup is normal is to check if every element in the subgroup commutes with every element in the larger group. If this is true, then the subgroup is normal. Another method is to use the definition of a normal subgroup and check if the subgroup is invariant under conjugation.
If a subgroup is normal, then the quotient group formed by dividing the larger group by the normal subgroup is a well-defined group. This relationship is known as the Fundamental Theorem of Homomorphisms and is a fundamental concept in group theory.
Yes, a group can have multiple normal subgroups. In fact, every group has at least two normal subgroups, the trivial subgroup containing only the identity element and the entire group itself. However, there are also groups with a larger number of normal subgroups.