Welcome to MAT301 - Groups and Symmetries
My office hour will resume this week on the usual
Friday schedule, i.e. after this lecture.
HW4 is due today, July 14, at 11:59pm.
Reminder - The First Isomorphism Theorem
Lemma
Let φ : G → G be a homomorphism. Then Ker(φ) ⊂ G
is a normal subgroup.
Theorem (First Isomorphism Theorem)
Let φ : G → G be a homomorphism. Consider a
mapping
φ : G/Ker(φ)→ Im(φ),
φ(gKer(φ)) = φ(g) ∈ Im(φ).
Then this is a well-defined group isomorphism. (usually
called the natural homomorphism or the quotient
map)
The Normalizer
Definition
Let G be a group, and let H be a subgroup. The
normailzer of H is a set
NG (H) = N(H) = {g ∈ G : gHg−1 = H}.
Theorem
Let H ⊂ G be a subgroup.
1. The normalizer is a subgroup of G .
2. H ⊆ N(H),
3. H is a normal subgroup of N(G ). Moreover, N(G )
is the largest subgroup of G with this property.
4. N(H) = G if and only if H is normal.
The Second Isomorphism Theorem
Theorem (Second Isomorphism Theorem)
Let A,B be subgroups of G , and assume that A is a
subgroup of N(B). Then the following statements hold:
1. AB is a subgroup of G
2. B is normal in AB , A ∩ B is normal in A,
3. AB/B ∼= A/A ∩ B .
The Third Isomorphism Theorem
Theorem (Third Isomorphism Theorem)
Let G be a group, and consider two normal subgroups
H ,K such that H is a subgroup of K . Then
1. K/H is a normal subgroup of G/H ,
2. (G/H)/(K/H) ∼= G/K .