Normal subgroup

Definition

A normal subgroup is a subgroup that is invariant under conjugation by any element of the larger group, meaning for a subgroup H of a group G, for every element g in G and every element h in H, the element gHg^{-1} is also in H. This property is crucial because it allows for the formation of quotient groups and is essential in the context of homomorphism theorems, which connect the structure of groups through their subgroups.

5 Must Know Facts For Your Next Test

1. For a subgroup H to be normal in G, it must hold that gHg^{-1} = H for all g in G.
2. Normal subgroups are the only subgroups that allow for the construction of quotient groups, which play a significant role in group theory.
3. Every subgroup of an abelian group is normal since all elements commute, making conjugation trivial.
4. The kernel of a homomorphism is always a normal subgroup of the original group, tying together concepts of structure and mapping between groups.
5. If N is a normal subgroup of G, then there exists an induced homomorphism from G to the quotient group G/N.

Review Questions

• How does the property of normality influence the structure and behavior of groups when considering quotient groups?
  • The property of normality is essential when creating quotient groups because it ensures that cosets formed from a normal subgroup are well-defined. If H is normal in G, then every left coset gH equals its corresponding right coset Hg. This uniformity allows us to treat these cosets as single entities, facilitating the formation of the quotient group G/H, which retains useful algebraic properties derived from G.
• Discuss how normal subgroups relate to homomorphisms and their kernels within group theory.
  • Normal subgroups are directly linked to homomorphisms since the kernel of any homomorphism between two groups is always a normal subgroup. This relationship highlights how we can understand group actions and mappings through the lens of substructures. The presence of normal subgroups allows us to use the First Isomorphism Theorem, which states that there exists an isomorphism between the quotient group formed by the kernel and the image of the homomorphism, showcasing how algebraic structures can mirror each other.
• Evaluate how identifying normal subgroups within a group can simplify understanding its overall structure and lead to deeper insights about its composition.
  • Identifying normal subgroups can significantly simplify understanding a group's structure because they reveal fundamental relationships between elements through cosets and quotient formations. By recognizing these subgroups, we can utilize tools like the Correspondence Theorem, which provides insights into how subgroups of normal subgroups relate to those of the parent group. This understanding can lead to identifying larger symmetries and simplifying complex calculations by reducing them to simpler quotient structures, thereby enhancing our grasp on the group's behavior and properties.

Related terms

Subgroup: A subgroup is a subset of a group that is itself a group under the same operation as the larger group.

Coset: A coset is a form of partitioning a group into equal-sized pieces, formed by multiplying all elements of a subgroup by a fixed element of the group.

Quotient Group: A quotient group is the set of cosets of a normal subgroup, which forms a new group with its own operation defined by the original group's operation.