A normal subgroup is a subgroup that is invariant under conjugation by any element of the group, meaning that for a subgroup H of a group G, for all elements g in G and h in H, the element gHg^{-1} is still in H. This property allows for the formation of quotient groups and is essential in understanding group structure and homomorphisms.
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Normal subgroups are crucial for defining quotient groups, which simplify the study of group properties.
Every subgroup of an abelian group is normal since the operation commutes.
The intersection of normal subgroups is also a normal subgroup.
The kernel of a homomorphism is always a normal subgroup of the domain group.
If H is a normal subgroup of G, then for every g in G, gHg^{-1} equals H.
Review Questions
How does the property of normality influence the relationship between a subgroup and its parent group?
The property of normality ensures that conjugation by any element from the parent group leaves the subgroup invariant. This means that if H is a normal subgroup of G, then for any element g in G and any element h in H, the element gHg^{-1} remains within H. This stability under conjugation is what allows for meaningful constructions like quotient groups, which reflect the structure of G while incorporating H.
Explain how normal subgroups relate to cosets and why this relationship is important in group theory.
Normal subgroups are directly related to cosets because they allow for the formation of well-defined left and right cosets that can be grouped together into quotient groups. When H is normal in G, all left cosets coincide with right cosets. This equality ensures that operations between cosets are well-defined, making it possible to work with quotient groups that reveal important structural properties about the original group G. This connection is fundamental when applying Lagrange's Theorem and analyzing group actions.
Analyze the implications of having a non-normal subgroup within a group and how it affects group homomorphisms.
If a subgroup is not normal within its parent group, it leads to complications in defining cosets and consequently quotient groups. Non-normality implies that conjugating elements can yield results outside the subgroup, thus making operations between such cosets undefined or inconsistent. This inconsistency carries over to homomorphisms as well; the kernel must be a normal subgroup to form valid quotient structures. The failure to be normal limits our ability to leverage powerful results like the First Isomorphism Theorem effectively.