Download Factor Group Computations, Simple Groups and Solvable Groups | MATH 5310 and more Study notes Abstract Algebra in PDF only on Docsity!
3.3. (III-15) FACTOR GROUP COMPUTATIONS, SIMPLE GROUPS, AND SOLVABLE GROUPS 45
3.3 (III-15) Factor Group Computations, Simple
Groups, and Solvable Groups
3.3.1 Factor Group Computations
Let N be a normal subgroup of G. The relations between the group G, the subgroup N , and the factor group G/N are shown in Figure 15.4. Think about the example that G = R^2 , N = {(0, y) | y ∈ R}. Then a coset a + N = {(a, y) | y ∈ R}.
There is no universal way to classify the factor group. However, we can solve some easy cases.
Ex 3.39. If a group G has the identity e, then G/{e} ' G.
Thm 3.40. A factor group of a cyclic group is cyclic.
Proof. 6 Refer to the proof in the textbook. We give another interpretation: Let G/N be a factor group of a cyclic group G. Then N is a normal subgroup. So N = ker(φ) for a group homomorphism φ : G → G′. Thus G/N ' φ[G]. If G is generated by a, then φ[G] is generated by φ(a). So φ[G] and G/N is cyclic.
Similarly,
Thm 3.41. A factor group of an abelian group is abelian.
Note that a factor group of a nonabelian group can also be abelian.
Ex 3.42 (Ex 15.4, p.145). When n > 2, the symmetric group Sn is nonabelian, but the factor group Sn/An ' Z 2 is abelian.
Thm 3.43. If Hi is a normal subgroup of Gi, then the factor group
( (^) n ∏
i=
Gi
( (^) n ∏
i=
Hi
∏^ n
i=
(Gi/Hi)
The theorem includes the case in Theorem 15.8 (p.147). You can try to construct the isomorphism explicitly.
(^6) 1st HW: 19, 20, 22, 28, 40
46 CHAPTER 3. HOMOMORPHISMS AND FACTOR GROUPS
3.3.2 Simple Groups
Def 3.44. A group G is simple if it is nontrivial and it has no proper non- trivial normal subgroups. That is, |G| > 1, and the only normal subgroups of G are {e} and G itself.
Thm 3.45. The alternative group An is simple for n ≥ 5.
The classification of all finite simple groups are done around 1980. It is a milestone in group theory.
Thm 3.46. Let φ : G → G′^ be a group homomorphism. If N is a normal subgroup of G, then φ[N ] is a normal subgroup of φ[G]. Also, if N ′^ is a normal subgroup of G′, then φ−^1 [N ′] is a normal subgroup of G.
3.3.3 The Center and Commutator Subgroups
Every group G has two important normal subgroups.
Def 3.47. The center of a group G is
Z(G) := {z ∈ G | zg = gz for all g ∈ G}.
It contains the elements that commutes with all elements of G.
Ex 3.48. The center Z(G) is a normal subgroup of G.
Def 3.49. The commutator subgroup of G is the group C generated by all elements of the set
{aba−^1 b−^1 | a, b ∈ G}.
We use C := [G, G] to represent the commutator subgroup.
Thm 3.50. The commutator subgroup C is a normal subgroup of G. More- over, if N is a normal subgroup of G. Then G/N is abelian if and only if C ≤ N ≤ G.
3.3.4 Solvable Groups
Other than simple group, another important type of groups is called the solvable group.
