Normal Subgroup

Definition

A normal subgroup is a subgroup that is invariant under conjugation by any element of the group, meaning that for any element in the group and any element in the subgroup, the result of conjugating the subgroup element remains in the subgroup. This property is essential because it allows for the formation of quotient groups, which helps in understanding the structure of larger groups through their simpler components. Normal subgroups play a key role in the study of group actions and algebraic groups, as they relate to the behavior of elements under these actions.

5 Must Know Facts For Your Next Test

1. Every group has at least two normal subgroups: the trivial subgroup containing just the identity and the group itself.
2. If a subgroup is normal, it can be used to define a quotient group, which simplifies the analysis of the original group.
3. A subgroup 'N' is normal in 'G' if and only if for all 'g' in 'G', the conjugate of any element 'n' in 'N' (i.e., 'gng^{-1}') is still in 'N'.
4. In abelian groups, every subgroup is normal because the group operation commutes, making conjugation trivial.
5. The intersection of two normal subgroups is also a normal subgroup, making them stable under various operations.

Review Questions

• How does the property of normality in a subgroup relate to the concept of quotient groups?
  • The property of normality is crucial for forming quotient groups because only normal subgroups allow for a well-defined set of cosets that can be grouped together to form another group. When a subgroup 'N' is normal in a group 'G', we can take the elements of 'G' and partition them into cosets of 'N'. This leads to a new structure where we can analyze properties and behavior without dealing with the complexity of 'G' directly.
• What are some examples of normal subgroups, and how do they illustrate the concept within different types of groups?
  • In cyclic groups, every subgroup is normal, showcasing that in abelian structures, normality is straightforward. For example, in the symmetric group S3, which includes permutations, the subgroup consisting of even permutations forms a normal subgroup because conjugating an even permutation by any permutation results in another even permutation. This illustrates how specific elements interact under group operations and highlights differences between abelian and non-abelian groups.
• Evaluate how normal subgroups influence group actions and provide an example demonstrating their significance.
  • Normal subgroups significantly influence group actions by determining how elements interact with each other during these actions. For instance, when considering a group acting on a set, if we have a normal subgroup, we can consider how this action simplifies through quotient structures. For example, if we take G = S4 acting on 4 objects, and let N be a normal subgroup like A4 (the alternating group), studying G/N allows us to understand how G acts on cosets rather than on all elements directly. This not only simplifies computations but also reveals deeper insights about symmetry and transformations.

Related terms

Subgroup:

A subgroup is a subset of a group that itself forms a group under the same operation.

Quotient Group: A quotient group is formed by partitioning a group into disjoint subsets (cosets) based on a normal subgroup, allowing for a simpler structure to be analyzed.

Conjugation: Conjugation is an operation in a group where an element is transformed by another element, specifically, for an element 'g' and 'h', it produces 'ghg^{-1}'.