%%visits: 2 They have a sense of invariance, thinking of normal, or orthogonal, it helps to understand that they are not affected by some sort of changing_the_basis seen in [[similar_matrices]]
normal_subgroups are where right coset and left cosets are the same.
[[conjugation]] makes way for normal_subgroups, which make the left coset equal to the right coset
normal_subgrou := When H ⊂ G is a normal_subgroup
if it is a subgroup and $h H, g G ghg^{-1} H := The set of
all conjugate elements are in the subgroup ## intuition
If G2 ⊲ G1 then G1/G2 forms a group
Proof: Let K = ker ϕ ⊂ G. We first show that K = ker (ϕ) is a subgroup. Closure: let k1, k2 ∈ K = ker ϕ then k1k2 ∈ ker ϕ as ϕ(k1k2) = ϕ(k1)ϕ(k2) = e′e′ = e therefore k1k2 ∈ ker ϕ. Associativity is inherited. Identity: e ∈ kerϕ as ϕ(e) = e′. Inverses For k ∈ ker ϕ then k−1 ∈ ker ϕ as ϕ(k−1) = ϕ(k)−1 = e′−1 = e′ Therefore k−1 ∈ ker ϕ so K is a subgroup.
We want to show that it is normal.Let K = ker ϕ
ϕ(gkg−1) = ϕ(g)ϕ(k)ϕ(g−1)
= ϕ(g)eϕ(g−1)
= ϕ(gg−1)
= e′
Consequently by this theorem, [[simple_group]] only have trivial [[homomorphism]]s.
Every normal_subgroup can be written as a kernel of a homomorphism
For every [[homomorphism]] there is a normal_subgroup. They are the same number.
[[special_linear_group]] ⊲ [[general_linear_group]]. PROOF, since the special_linear_group is the kernel of the determinant homomorphism det : GL(n, ℂ) → ℂ× then it is a normal_subgroup by the [[homomorphism_theorem]] ## rigour ## exam clinic ## examples and non-examples ## resources tags :math:groups_and_symmetries:
ormality of a subgroup allows us to have multiplication on [[coset]] space.