1.5. NORMALITY, QUOTIENT GROUPS, AND HOMOMORPHISMS
1.5
17
Normality, Quotient Groups, and Homomorphisms
Let N < G. If the left congruence modulo N relation defines a group structure over G/N , the
set of left cosets of N in G, then we must have (bN )(aN ) = baN for a, b ∈ G. In other words,
N a = aN for a ∈ G (left coset=right coset).
Def. A subgroup N of a group G is called a normal subgroup of G, written as N C G, if
one of the following equivalent conditions holds:
1. aN = N a for a ∈ G;
2. aN a−1 = N for a ∈ G;
3. aN a−1 ⊂ N for a ∈ G.
(proof)
Ex.
1. Every subgroup of an abelian group is normal.
2. Every subgroup of G of index 2 is a normal subgroup.
3. In S3 , the subgroup e, 12 23 31 , 13 21 32 is a normal subgroup.
Thm 1.14. If N C G, then the set G/N of all (left) cosets of N in G is a group of order
[G : N ] under the operation (aN )(bN ) := abN .
The group G/N is called the quotient group or factor group of G by N .
Thm 1.15. Let K < G and N C G. Then
1. (N ∩ K) C K;
2. N C (N ∨ K) = N K = KN ;
Therefore, if one of the subgroups K and N of G is normal, then KN is a subgroup of G.
Normal subgroups of a group G are related to the kernels of homomorphisms from G to
another subgroups.
Thm 1.16. Let f : G → H be a homomorphism of groups. Then Ker f C G. Conversely,
if N C G, then there exists the canonical projection π : G → G/N by a 7→ aN such that
Ker π = N .
18
CHAPTER 1. GROUPS
Thm 1.17 (First Isomorphism Theorem). If f : G → H is a homomorphism of groups, then
∼
f induces an isomorphism fe : G/Ker f → Im f given by a(Ker f ) 7→ f (a) for a ∈ G.
G
f
π
G/Ker f
fe
/H
O
?
ι
/ Im f
(proof)
Prop 1.18. Let f : G → H be a homomorphism of groups. N C G, M C H, and f (N ) < M .
Then f induces a homomorphism fe : G/N → H/M given by aN 7→ f (a)M .
Thm 1.19 (Second Isomorphism Theorem). If K < G and N CG, then K/(K ∩N ) ' KN/N .
(proof)
Thm 1.20 (Third Isomorphism Theorem). If K < H < G, K C G and H C G, then K C H
and G/H ' (G/K)/(H/K).
Thm 1.21. If f : G → H is an epimorphism of groups. Then there is a one-to-one correspondence between the set of all subgroups K of G which contain Ker f and the set of all subgroups
of H, given by K 7→ f (K).
In particular, if N C G, then there is a one-to-one correspondence between the set of all
subgroups of G that contain N and the set of all subgroups of G/N .