THE FACTOR GROUP

Chapter 7 THE FACTOR GROUP

The factor group is a group that contains cosets of a subgroup. De- velopment of a factor group can be started by a homomorphism or a normal subgroup.

7.1 Factor Group Development by a Homomor- phism

Theorem 7.1.1 ′ Let φ : G → G is a homomorphism with Ker(φ) = H. Then R/H = {a ∗ H|a ∈ R} is a group with the binary operation:

(aH)(bH) = (ab)H

Dan the mapping µ : G/H → φ(G) defined by µ(a∗H) = φ(a), is an isomorphism. Example 7.1.1 The mapping φ : Z → Z n defined by φ(m) = r, where r is the

reminder when m is divided by n, is a homomorphism. Since Ker(φ) = nZ, hence Z/nZ is a group that isomorphs to Z n .

7.2 Factor Group Development by Normal Sub- group

Theorem 7.2.1 Let H be a subgroup of group G. Multiplications of cosets of H, that is defined by (aH)(bH) = (ab)H is well-defined if and only if aH = Ha,

∀a ∈ G Corollary 7.2.1 Let H be a normal subgroup of a group G. Then G/H =

{aH|a ∈ G} is a group with binary operation:

(aH)(bH) = (ab)H

Definition 7.2.1 The group G/H as described in the above corollary is Quo- tient Group

G modulo H Example 7.2.1 If Z is an abelian group, then nZ is a normal subgroup of Z,

therefore it can be form a factor group Z/nZ without introducing a homomor- phism.

Some conditions below are equivalent characteristics on a normal subgroup H in a group G.

1. ghg −1 ∈ H, ∀g ∈ G and h ∈ H.

2. gHg −1 = H, ∀g ∈ G.

3. gH = Hg, ∀g ∈ G. Show how that three conditions can be derived from the normal subgroup defin-

ition? To do this, use one of the characteristics above. Definition 7.2.2 An isomorphism φ : G → G is called automorphism in G.

An automorphism I −1

g : G → G where I g (x) = gxg is called inner automor- phism of G by g.

7.3 The Fundamental Homomorphism Theorem

Theorem 7.3.1 Let H be a factor group of a group G. Then φ : G → G/H defined by φ(a) = aH, is a homomorphism with Ker(φ) = H.

Theorem 7.3.2 The Fundamental Homomorphism Theorem. Let φ : G→G ′

be a homomorphism with Ker(φ) = H. Then φ(G) is a group, and the mapping µ : G/H → φ(G), defined by µ(aH) = φ(a), is an isomorphism. If

γ : G → G/H is a homomorphism that is defined by γ(a) = aH, then ∀a ∈ G, φ(a) = µγ(a).

7.4 Exercises on the Factor Group

1. Compute the order of each factor group below. (a) Z 6 /<3>

(b) (Z 4 ×Z 2 )/ < (2, 1) > (c) (Z 2 ×Z 4 )/ < (1, 1) >

(d) (Z 2 ×S 3 )/ < (1, ρ 1 )> (e) (Z 4 ×Z 12 )/(< 2 > × < 2 >) (f) (Z 3 ×Z 5 )/({0} × Z 5 ) (g) (Z 12 ×Z 18 )/ < (4, 3) > (h) (Z 11 ×Z 15 )/ < (1, 1) >

2. Compute the order of each element of the following factor group.

(a) 5+ < 4 > in Z 12 / < 4 >. (b) 26+ < 12 > in Z 60 / < 12 >. (c) (2, 1)+ < (1, 1) > in (Z 3 ×Z 6 )/ < (1, 1) >. (d) (3, 1)+ < (1, 1) > in (Z 4 ×Z 4 )/ < (1, 1) >. (e) (3, 1)+ < (0, 2) > in (Z 4 ×Z 8 )/ < (0, 2) >. (f) (3, 3)+ < (1, 2) > in (Z 4 ×Z 8 )/ < (1, 2) >. (g) (2, 0)+ < (4, 4) > in (Z 6 ×Z 8 )/ < (4, 4) >.

3. Determine true or false each statement below and give the reason.

(a) A factor group G/N can be developed if and only if N is a normal subgroup of G.

(b) Every subgroup of an abelian group is normal. (c) An automorphism inner in an abelian group must be an identity func-

tion. (d) Every factor group of a finite group is also finite. (e) Every factor group of an abelian group is also abelian.

(f) Every factor group of a non abelian group is also non abelian. (g) Z/nZ is a cyclic group of ordern.

4. Let H be a normal subgroup of a group G, and let m = (G : H). Show that a m ∈ H, for every a ∈ G!

5. Show that intersection of normal subgroups is also a normal subgroup!

6. Show that if a finite group G has exactly one subgroup H of certain order, then H is also normal subgroup of G!

7. Show that if H and N are subgroup of group G and N is a normal subgroup, then H ∩ N is normal in H! Show using an example that H ∩ N is not necessary a normal of G!