Today’s Trivia
Welcome to MAT301 - Groups and Symmetries
My office hour will resume this week on the usual
Friday schedule.
Tutorials resume this week.
HW4 was posted last week and it is due on Friday,
July 14, at 11:59pm.
Cauchy’s Theorem for Finite Abelian Groups
Theorem
If G is a finite Abelian group and p is a prime divisor of
|G |, then there is an element a ∈ G of order p.
Notice the connection with Lagrange’s theorem.
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 .