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§ 3.7 Homomorphisms

 
Definition 3.7.1. Let G₁ and G₂ be groups, and let µ : G₁ -> G₂ be a function. Then µ is said to be a group homomorphism if

µ(ab) = µ(a) µ(b)

 
Example 3.7.1. (Exponential functions for groups) Let G be any group, and let a be any element of G. Define µ : Z -> G by µ(n) = aⁿ, for all n in Z. This is a group homomorphism from Z to G.
If G is abelian, with its operation denoted additively, then we define µ : Z -> G by µ(n) = na.

 
Example 3.7.2. (Linear transformations) Let V and W be vector spaces. Since any vector space is an abelian group under vector addition, any linear transformation between vector spaces is a group homomorphism.

 
Proposition 3.7.2. If µ : G₁ -> G₂ is a group homomorphism, then

(a) µ(e) = e;

(b) (µ(a))^-1 = µ(a^-1) for all a in G₁;

(c) for any integer n and any a in G₁, we have µ(aⁿ) = (µ(a))ⁿ;

(d) if a is in G₁ and a has order n, then the order of µ(a) in G₂ is a divisor of n.

 
Example 3.7.4. (Homomorphisms defined on cyclic groups) Let C be a cyclic group, denoted multiplicatively, with generator a. If µ : C -> G is any group homomorphism, and µ(a) = g, then the formula must hold. Since every element of C is of the form a^m for some integer m, this means that µ is completely determined by its value on a.
If C is infinite, then for an element g of any group G, the formula µ(a^m) = g^m defines a homomorphism.
If |C|=n and g is any element of G whose order is a divisor of n, then the formula µ(a^m) = g^m defines a homomorphism.

 
Example 3.7.5. (Homomorphisms from Z_n to Z_k) Any homomorphism µ : Z_n -> Z_k is completely determined by µ([1]_n), and this must be an element [m]_k of Z_k whose order is a divisor of n. Then the formula µ([x]_n) = [mx]_k, for all [x]_n in Z_n, defines a homomorphism. Furthermore, every homomorphism from Z_n into Z_k must be of this form. The image µ(Z_n) is the cyclic subgroup generated by [m]_k.

 
Definition 3.7.3. Let µ : G₁ -> G₂ be a group homomorphism. Then

{ x in G₁ | µ(x) = e }

 
Proposition 3.7.4. Let µ : G₁ -> G₂ be a group homomorphism, with K = ker(µ).

(a) K is a normal subgroup of G.

(b) The homomorphism µ is one-to-one if and only if K = {e}.

 
Definition 3.7.5. A subgroup H of the group G is called a normal subgroup if

ghg^-1 belongs to H

 
Proposition 3.7.6. Let µ : G₁ -> G₂ be a group homomorphism.

(a) If H₁ is a subgroup of G₁, then µ(H₁) is a subgroup of G₂.
If µ is onto and H₁ is normal in G₁, then µ(H₁) is normal in G₂.

(b) If H₂ is a subgroup of G₂, then

µ^-1 (H₂) = { x in G₁ | µ(x) is in H₂ }

is a subgroup of G₁.
If H₂ is normal in G₂, then µ^-1(H₂) is normal in G₁.

 
Let µ : G₁ -> G₂ be a group homomorphism. The function µ determines an equivalence relation of G₁ by setting a ~ b if µ(a) = µ(b). The notation G₁/µ is used for the set of equivalence classes of this relation.

 
Proposition 3.7.7. Let µ : G₁ -> G₂ be a group homomorphism. Then multiplication of equivalence classes in G₁/µ is well-defined, and G₁/µ is a group under this multiplication. The natural mapping : G₁ -> G₁/µ defined by (x) = [x]_µ is a group homomorphism.

 
Theorem 3.7.8. Let µ : G₁ -> G₂ be a group homomorphism. Then G₁/µ is isomorphic to µ(G₁).

 
Proposition 3.7.9. Let µ : G₁ -> G₂ be a group homomorphism, and let a,b belong to G₁. The following conditions are equivalent:

(1) µ(a) = µ(b)

(2) ab^-1 belongs to ker(µ);

(3) a = kb for some k in ker(µ);

(4) b^-1a belongs to ker(µ);

(5) a = bk for some k in ker(µ);

§ 3.7 Homomorphisms: Solved problems

Theorem 3.7.8 is the most important result, but Proposition 3.7.6 is also useful, since for any group homomorphism µ : G₁ -> G₂ it describes the connections between subgroups of G₁ and subgroups of G₂. Examples 3.7.4 and 3.7.5 are important, because they give a complete description of all group homomorphisms between two cyclic groups.

17. Find all group homomorphisms from Z₄ into Z₁₀.     Solution

18. (a) Find the formulas for all group homomorphisms from Z₁₈ into Z₃₀.
    (b) Choose one of the nonzero formulas in part (a), and for this formula find the kernel and image, and show how elements of the image correspond to cosets of the kernel.
    Solution

19. (a) Show that Z₇^× is cyclic, with generator [3]₇.
    (b) Show that Z₁₇^× is cyclic, with generator [3]₁₇.
    (c) Completely determine all group homomorphisms from Z₁₇^× into Z₇^×.
    Solution

20. Define µ : Z₄ × Z₆ -> Z₄ × Z₃ by

µ ([x]₄,[y]₆) = ([x+2y]₄,[y]₃).

21. Let n and m be positive integers, such that m is a divisor of n. Show that
µ : Z_n^× -> Z_m^× defined by

µ ( [x]_n ) = [x]_m,     for all [x]_n in Z_n^×,

22. For the group homomorphism µ : Z₃₆^× -> Z₁₂^× defined by

µ ( [x]₃₆ ) = [x]₁₂,     for all [x]₃₆ in Z₃₆^×,

23. Let G, G₁, and G₂ be groups. Let µ₁ : G -> G₁ and µ₂ : G -> G₂ be group homomorphisms. Prove that
µ : G -> G₁ × G₂ defined by

µ (x) = (µ₁ (x), µ₂ (x)),     for all x in G,

24. Let p and q be different odd primes. Prove that Z_pq^× is isomorphic to the direct product Z_p^× × Z_q^× .     Solution

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