A normal subgroup is a special type of subgroup within a group in abstract algebra. It is defined as a subgroup that remains invariant under conjugation by any element of the larger group. This means that if you take any element from the normal subgroup and any element from the group, the result of conjugating the subgroup element by the group element will still be in the subgroup. Normal subgroups are important because they allow for the construction of quotient groups, which are essential in group theory. A common example of a normal subgroup is the center of a group, which consists of elements that commute with every element in the group. Other related concepts include group homomorphisms and simple groups.