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Chapter 3
Homomorphisms and Factor
Groups
3.1 (III-13) Homomorphisms
3.1.1 Structure-Relating Maps
Isomorphism is a bijection between groups that satisfies the homomor- phism property. If we have an isomorphism φ : G → G′^ and we know the structure of G, then the structure of G′^ is just the same as G. More powerful tools are needed to study the structures of groups. Def 3.1. A Homomorphism is a map between groups (not necessary a bijection) that satisfies the homomorphism property: φ : G → G′, φ(ab) = φ(a)φ(b) for all a, b ∈ G. How homomorphisms preserve group structures? See the examples. Ex 3.2. Let φ : G → G′^ be defined by φ(g) = e′^ for g ∈ G, where e′^ is the identity of G′. Then φ is a trivial homomorphism. Ex 3.3. If H ≤ G, then τ : H → G defined by τ (h) := h for h ∈ H is a group homomorphism. Ex 3.4. A linear transformation φ : V → W between real vector spaces V and W is a homomorphism of abelian groups V and W. Ex 3.5 (Ex 13.2, p.126). 1 Let φ : G → G′^ be a group homomorphism of G onto G′^ (i.e. surjective). If G is abelian then so is G′. (^1) 1st HW: 2, 4, 6, 8, 10, 49 35
36 CHAPTER 3. HOMOMORPHISMS AND FACTOR GROUPS
Ex 3.6. Let φ : G → G′^ be a group homomorphism, then φ(g−^1 ) = φ(g)−^1. Ex 3.7 (Ex 13.3, p.126). Let φ : Sn → Z 2 be defined by
φ(σ) :=
{ 0 , if σ is an even permutation, 1 , if σ is an odd permutation. Then φ is a homomorphism. Ex 3.8 (Ex 13.4, p.126, Evaluation Homomorphism). Let F be the additive group of all functions mapping R into R. For c ∈ R, the map φc : F → R defined by φc(f ) := f (c) is a homomorphism between 〈F, +〉 and 〈R, +〉, called the evaluation homomorphism (at c). Ex 3.9 (det). The determinant map of nonsingular n × n matrices det : GL(n, R) → R∗ is a homomorphism between 〈GL(n, R), ·〉 and 〈R∗, ·〉. Ex 3.10 (Ex 13.8 p.127, Hw 7 p.133). Let G = G 1 × G 2 × · · · × Gn be the direct product of groups. Let i = 1, 2 , · · · , n.
- The projection map πi : G → Gi defined by πi((g 1 , g 2 , · · · , gn)) := gi is a group homomorphism.
- The injection map τi : Gi → G defined by τi(gi) := (e 1 , e 2 , · · · , ei− 1 , gi, ei+1, · · · , en) where ej is the identity of Gj , is a group homomorphism. Ex 3.11 (Ex 13.7, p.127). Let r ∈ Z. The map φr : Z → Z, defined by φr(x) = rx for x ∈ Z, is a homomorphism. φ 0 is the trivial map; φ 1 is the identity map; In general, φr maps Z onto the subgroup rZ ≤ Z. Ex 3.12 (Ex 13.10, p.127). Let n ∈ Z+. Let φn : Z → Zn be defined by φ(x) := x mod n, where 0 ≤ x mod n < n is the remainder of x divided by n. Then φn is a homomorphism. Ex 3.13. Let m, n ∈ Z+.
- The map φ : Zmn → Zn, defined by φ(x) := x mod n for x ∈ Zmn, is a homomorphism.
- The map ψ : Zn → Zmn, defined by φ(x) := mx for x ∈ Zn, is a homomorphism.
38 CHAPTER 3. HOMOMORPHISMS AND FACTOR GROUPS
Thm 3.17. 3 Let φ : G → G′^ be a group homomorphism. Let H := ker(φ). Then for every a ∈ G, the inverse image of φ(a) is
φ−^1 [{φ(a)}] := {x ∈ G | φ(x) = φ(a)} = aH = Ha.
Ex 3.18 (Ex 13.17 p.131). Differentiation operation is a group homomor- phism. The kernel and the inverse image of x^2. Cor 3.19. A group homomorphism φ : G → G′^ is a one-to-one map if and only if ker(φ) = {e}. (Recall that φ is onto if and only if φ[G] = G′). Def 3.20. A subgroup H of G is normal if gH = Hg for all g ∈ G. Ex 3.21. If G is abelian, then every subgroup H of G is normal. Cor 3.22. If φ : G → G′^ is a group homomorphism, then ker(φ) is a normal subgroup. Proof. Let H := ker(φ). By Thm 3.17, aH = Ha for all a ∈ G. So H is normal.
3.1.3 Homework, III-13, p.133-p.
1st: 2, 4, 6, 8, 10, 49 2nd: 18, 21, 29, 44, 45 3rd: 32 , 35, 36, 51, 52, opt: 46, 50, 55.
(^3) 2nd HW: 18, 21, 29, 44, 45
