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Overview / review of cosets, factor groups, and homomorphisms for MATH 421, from Abstract Algebra, 3rd Ed., Sections 3.7 – 3.
Cosets and normal subgroups
Definition 3.8.3. Let H be a subgroup of the group G, and let a ∈ G. The set
aH = {x ∈ G | x = ah for some h ∈ H}
is called the left coset of H in G determined by a. Similarly, the right coset of H in G determined by a is the set Ha = {x ∈ G | x = ha for some h ∈ H}. The number of left cosets of H in G is called the index of H in G, and is denoted by [G : H].
Proposition 3.8.1. Let H be a subgroup of the group G, and let a, b ∈ G. Then the following conditions are equivalent: (1) bH = aH; (2) bH ⊆ aH; (3) b ∈ aH; (4) a−^1 b ∈ H.
A result similar to Proposition 3.8.1 holds for right cosets. Let H be a subgroup of the group G, and let a, b ∈ G. Then the following conditions are equivalent: (1) Ha = Hb; (2) Ha ⊆ Hb; (3) a ∈ Hb; (4) ab−^1 ∈ H; (5) ba−^1 ∈ H; (6) b ∈ Ha; (7) Hb ⊆ Ha. The index of H in G could also be defined as the number of right cosets of H in G, since there is a one-to-one correspondence between left cosets and right cosets.
Definition 3.7.5. A subgroup H of the group G is called a normal subgroup if ghg−^1 ∈ H for all h ∈ H and g ∈ G.
Proposition 3.8.8. Let H be a subgroup of the group G. The following conditions are equivalent: (1) H is a normal subgroup of G; (2) aH = Ha for all a ∈ G; (3) for all a, b ∈ G, abH is the set theoretic product (aH)(bH); (4) for all a, b ∈ G, ab−^1 ∈ H if and only if a−^1 b ∈ H.
Example 3.8.8. Any subgroup of index 2 is normal.
Factor groups
Proposition 3.8.4. Let N be a normal subgroup of G, and let a, b, c, d ∈ G. If aN = cN and bN = dN , then abN = cdN.
Theorem 3.8.5. If N is a normal subgroup of G, then the set of left cosets of N forms a group under the coset multiplication given by aN bN = abN for all a, b ∈ G.
Definition 3.8.5. If N is a normal subgroup of G, then the group of left cosets of N in G is called the factor group of G determined by N. It will be denoted by G/N.
Example 3.8.5. Let N be a normal subgroup of G. If a ∈ G, then the order of aN is the smallest positive integer n such that an^ ∈ N.
Group homomorphisms
Definition 3.7.1. Let G 1 and G 2 be groups, and let φ : G 1 → G 2 be a function. Then φ is said to be a group homomorphism if φ(ab) = φ(a)φ(b) for all a, b ∈ G 1.
Example 3.7.3. (Exponential functions for groups) Let G be a group, and let a be any element of G. Define φ : Z → G by φ(n) = an, for all n ∈ Z. This is a group homomorphism from Z to G. If G is abelian, with its operation denoted additively, then we define φ : Z → G by φ(n) = na.
Example 3.7.4. (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 φ : G 1 → G 2 is a group homomorphism, then (a) φ(e) = e; (b) (φ(a))−^1 = φ(a−^1 ) for all a ∈ G 1 ; (c) for any integer n and any a ∈ G 1 , φ(an) = (φ(a))n; (d) if a ∈ G 1 and a has order n, then the order of φ(a) in G 2 is a divisor of n.
Example 3.7.6. (Homomorphisms defined on cyclic groups) Let C be a cyclic group, denoted multiplicatively, with generator a. If φ : C → G is any group homomorphism, and φ(a) = g, then the formula φ(am) = gm must hold. Since every element of C is of the form am^ for some m ∈ Z, this means that φ is completely determined by its value on a. If C is infinite, then for an element g of any group G, the formula φ(am) = gm^ defines a homomorphism. If |C| = n and g is any element of G whose order is a divisor of n, then the formula φ(am) = gm^ defines a homomorphism.
Example 3.7.7. (Homomorphisms from Zn to Zk) Any homomorphism φ : Zn → Zk is completely determined by φ([1]n), and this must be an element [m]k of Zk whose order is a divisor of n. Then the formula φ([x]n) = [mx]k, for all [x]n ∈ Zn, defines a homomorphism. Furthermore, every homomorphism from Zn into Zk must be of this form. The image φ(Zn) is the cyclic subgroup generated by [m]k.
Definition 3.7.3 Let φ : G 1 → G 2 be a group homomorphism. Then {x ∈ G 1 | φ(x) = e} is called the kernel of φ, and is denoted by ker(φ).
Proposition 3.7.4 Let φ : G 1 → G 2 be a group homomorphism, with K = ker(φ). (a) K is a subgroup of G 1 such that gkg−^1 ∈ K for all k ∈ K and g ∈ G 1. (b) The homomorphism φ is one-to-one if and only if K = {e}.
Proposition 3.7.6 Let φ : G 1 → G 2 be a group homomorphism. (a) If H 1 is a subgroup of G 1 , then φ(H 1 ) is a subgroup of G 2. If φ is onto and H 1 is normal in G 1 , then φ(H 1 ) is normal in G 2. (b) If H 2 is a subgroup of G 2 , then φ−^1 (H 2 ) = {x ∈ G 1 | φ(x) ∈ H 2 } is a subgroup of G 1. If H 2 is a normal in G 2 , then φ−^1 (H 2 ) is normal in G 1.
Proposition 3.8.7. Let N be a normal subgroup of G. (a) The natural projection mapping π : G → G/N defined by π(x) = xN , for all x ∈ G, is a homomor- phism, and ker(π) = N. (b) There is a one-to-one correspondence between subgroups of G/N and subgroups H of G with H ⊇ N. Under this correspondence, normal subgroups correspond to normal subgroups.
Example 3.8.10. Zn/mZn ∼= Zm if m|n.
Theorem 3.8.9. [Fundamental Homomorphism Theorem] Let G 1 , G 2 be groups. If φ : G 1 → G 2 is a homomorphism with K = ker(φ), then G 1 /K ∼= φ(G 1 ).
Definition 3.8.10. The group G is called a simple group if it has no proper nontrivial normal subgroups.
