1.2. HOMOMORPHISMS AND SUBGROUPS 11 1.2 Homomorphisms and Subgroups From now on, unless otherwise specified, we assume that G is a group. Def. Let G be a group and H a subset of G. If H is a group under the product in G, then H is said to be a subgroup of G, denoted by H < G. A group G always has subgroup G and trivial subgroup feg . The other subgroups of G are called proper subgroups of G. Ex. p 1. The additive groups Z < Q < Q[ 2] < R < C. 2. The permutations of f1; 2; ··· ; ng in Sn that leave n fixed form a subgroup isomorphic to Sn−1. Thm 1.3. A nonempty subset H of a group G is a subgroup of G if and only if ab−1 2 H for all a; b 2 H. T Cor 1.4. Let fHi j i 2 Ig be a nonempty family of subgroups of a group G. Then i2I Hi is a subgroup of G. Def. Let G be a subgroup and X a subset of G. Let fHi j i 2 Ig be the family of all subgroups T of G that contain X. Then i2I Hi is called the subgroup of G generated by X and denoted hXi. n1 n2 nt Remark. We have hXi = fa1 a2 ··· at j ai 2 X; ni 2 Zg. In particular, for a 2 G, n hai = fa j n 2 Zg is called a cyclic subgroup of G. Let H and K be subgroups of G. We use H _ K to denote the subgroup generated by H [ K. It is called the join of H and K. Def. Let G and H both be semigroups/monoids/groups. A function f : G ! H is a hmo- morphism provided f(ab) = f(a)f(b) for all a; b 2 G: Remark. 1. ab is the binary operation in G, and f(a)f(b) is the binary operation in H. The homomorphism f is called if f is monomorphism: injective epimorphism: surjective 2. isomorphism: bijective endomorphism: from G to G automorphism: a bijection from G to G 12 CHAPTER 1. GROUPS Ex. 1. f : Z ! Zm by x 7! x¯ := (x mod m) is a group epimorphism. 2. Let G and H be groups. The function ι1 : G ! G × H given by g 7! (g; eH ) is a group monomorphism. The function π1 : G × H ! G given by (g; h) 7! g is a group epimorphism. 3. Let Bn be the group of all nonsingular n × n upper triangular matrices, and Tn the group of all nonsingular n × n diagonal matrices. The function f : Bn ! Tn defined by a 7! f(a), where f(a) is the diagonal matrix of a, is a group epimorphism. The function g : Tn ! Bn defined by inclusion is also a group monomorphism. Def. Let f : G ! H be a group homomorphism. • The kernel of f is Ker f = fa 2 G j f(a) = eH g. It is a (normal) subgroup of G. • If A < G, then f(A) < H, where f(A) = ff(a) j a 2 Gg is the image of A under f. • If B < H, then f −1(B) < G, where f −1(B) = fa 2 G j f(a) 2 Bg is the inverse image of B under f. Remark. 1. A homomorphism f is a monomorphism if and only if Ker f = feg; f is an epimorphism if and only if f(G) = H. 2. For a 2 G, the inverse image of f(a) 2 H is f −1(f(a)) = a · Ker f = fan j n 2 Ker fg:.