7) Normal Subgroups and Factor Groups Flashcards
What is a normal subgroup
Let G be a group. A subgroup H ≤ G is called normal in G, if for all h ∈ H and all g ∈ G, we have ghg−1 ∈ H, in other words, if every conjugate of an element of H is again in H
What characterises a normal subgroup in terms of conjugacy classes
Normal subgroups are subgroups that are unions of conjugacy classes
Notice that H ≤ G if and only if H ≤ G and h^G ⊆ H for all h ∈ H. That is H ≤ G if and only if H ≤ G and H is the union of some of the conjugacy classes of G
What properties do all subgroups of index 2 have
Any subgroup of index 2 is normal
Let G be a group and H ≤ G with [G : H] = 2. Then H ≤ G
Describe the proof that any subgroup of index 2 is normal
What property do kernels have in terms of them being a subgroup
Kernels are normal subgroups
Describe the proof that Kernels are normal subgroups
What is the set of cosets of normal subgroups N in G
What does it mean if multiplication of cosets of a normal subgroup is well defined
What property do cosets of a normal subgroup have under coset multiplication
Cosets of a normal subgroup form a group under coset multiplication
Describe the proof that cosets of a normal subgroup form a group under coset multiplication
What is a Factor group
Let G be a group and N be a normal subgroup of G. The group G/N with operation * is called the factor group (or quotient group) of G by N
What is a homomorphism from G to G/N
What is the kernel of the homomorphism from G to G/N
Ker ν = N
Describe the proof that the homomorphism from G to G/N is a homomorphism
What is the natural homomorphism
The homomorphism ν : G → G/N is called the natural homomorphism from G to G/N
What is the First Isomorphism Theorem
How is Z/nZ isomorphic to Zn (where n is a natural number)
