Normal subgroups and quotient groups are key concepts in group theory. They help us understand group structure and relationships between groups. Normal subgroups are special because they allow us to create quotient groups, which simplify our study of the original group.
The first isomorphism theorem is a powerful tool for problem-solving in group theory. It connects homomorphisms, normal subgroups, and quotient groups, helping us classify groups and prove important relationships between different group structures.
Normal subgroups and their significance
Definition and properties of normal subgroups
- A subgroup H of a group G is normal if and only if gH = Hg for all g in G, where gH = {gh | h in H} and Hg = {hg | h in H}
- Equivalently, a subgroup H is normal if and only if gHg⁻¹ ⊆ H for all g in G, or gHg⁻¹ = H for all g in G
- Normal subgroups are invariant under conjugation by elements of the group G
- The trivial subgroup {e} and the entire group G are always normal subgroups of G
Importance of normal subgroups
- Normal subgroups allow the construction of quotient groups, which can simplify the study of the original group
- The kernel of a group homomorphism is always a normal subgroup, and the image of a group homomorphism is isomorphic to the quotient group of the domain by the kernel
- Normal subgroups play a crucial role in the classification of groups up to isomorphism
- Many important subgroups, such as the center of a group, the commutator subgroup, and the subgroup generated by the nth powers of elements in a group, are normal subgroups
Quotient groups and their structure
Construction of quotient groups
- Given a normal subgroup N of a group G, the quotient group (or factor group) G/N is the set of all left cosets of N in G, with the operation (aN)(bN) = (ab)N for a, b in G
- The elements of the quotient group are the cosets of the normal subgroup, and the group operation is well-defined because the subgroup is normal
- The canonical projection map π: G → G/N defined by π(g) = gN is a surjective group homomorphism with kernel N
Properties of quotient groups
- The order of the quotient group G/N is equal to the index of N in G, denoted [G:N], which is the number of distinct left (or right) cosets of N in G
- The quotient group G/N is abelian if and only if N contains the commutator subgroup [G, G]
- The quotient group G/Z(G), where Z(G) is the center of G, is isomorphic to the inner automorphism group Inn(G) of G
- The quotient group G/[G, G], where [G, G] is the commutator subgroup of G, is isomorphic to the abelianization of G, which is the largest abelian quotient of G
Isomorphism theorem for problem-solving
First isomorphism theorem
- The first isomorphism theorem states that if φ: G → H is a group homomorphism, then the kernel of φ, denoted ker(φ), is a normal subgroup of G, and the image of φ, denoted im(φ), is isomorphic to the quotient group G/ker(φ)
- The first isomorphism theorem can be used to prove that any surjective homomorphism φ: G → H induces an isomorphism between G/ker(φ) and H
- The first isomorphism theorem can be applied to prove the fundamental theorem of homomorphisms, which states that for any group homomorphism φ: G → H, there exists a unique group homomorphism ψ: G/ker(φ) → H such that φ = ψ ∘ π, where π is the canonical projection map
Applications of the first isomorphism theorem
- The first isomorphism theorem can be used to classify groups up to isomorphism by studying their quotient groups
- The first isomorphism theorem can be applied to solve problems involving the structure of groups and their homomorphisms
- The first isomorphism theorem can be used to prove the existence of certain subgroups or quotient groups (e.g., the existence of the abelianization of a group)
- The first isomorphism theorem can be used to establish relationships between different groups and their properties (e.g., the relationship between the center and the inner automorphism group of a group)
Theorems involving normal subgroups and quotient groups
Theorems for proving normality
- To prove that a subgroup H of a group G is normal, it suffices to show that gHg⁻¹ ⊆ H for all g in G, or equivalently, that gHg⁻¹ = H for all g in G
- The normalizer of a subgroup H in G, denoted N_G(H), is the largest subgroup of G in which H is normal
- A subgroup H of a group G is normal if and only if its left cosets and right cosets coincide, i.e., gH = Hg for all g in G
Correspondence and isomorphism theorems
- The correspondence theorem establishes a bijection between the subgroups of G containing a normal subgroup N and the subgroups of the quotient group G/N
- The third isomorphism theorem states that if H and N are normal subgroups of G with N ⊆ H, then (G/N)/(H/N) ≅ G/H, where H/N is a normal subgroup of G/N
- The second isomorphism theorem (or diamond isomorphism theorem) states that if H and K are subgroups of G with K normal, then HK is a subgroup of G, H ∩ K is a normal subgroup of H, and H/(H ∩ K) ≅ HK/K