IV. Homomorphisms and Normal Subgroups

Question 1

Definition
Conjugate.
Normal subgroup.

Question 2

Lemma IV.1.3
Equivalences to Normal subgroup.

Question 3

Lemma IV.1.8
If H subgroup of index 2.

Question 4

Definition
The set of left cosets.
If H is normal?

Question 5

Lemma IV.2.1
If N is Normal, then group operation on quotient group is well-defined.

Proof

Question 6

Lemma IV.2.2
If N Normal, then G/N

Question 7

Definition
Group homomorphism.

Question 8

Definition
Group isomorphism.
Kernel, image.

Question 9

Theorem IV.5.1
How do Ker and Im relate to G and H?

Question 10

Lemma IV.5.4
If \phi is a homom., \phi is injective iff.

Question 11

Theorem IV.6.1
The First Isomorphism Theorem

Question 12

Definition
sign(\sigma) is a homomorphism…

Question 13

Theorem IV.7.2
A_n and S_n

Question 14

Lemma IV.8.1
If G, H are cyclic of order…

Question 15

Lemma IV.8.4
If p prime, then any group of order p…

Question 16

Definition
Direct product.

Question 17

Lemma IV.8.5
GxH is a group…

Question 18

Theorem IV.8.6
The only groups of order 4…