1.2 Homomorphisms and Subgroups

Home , Epimorphism, Group homomorphism, Image (category theory), Image (mathematics), Monomorphism

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 {e} . The other subgroups of G are called proper subgroups of G. Ex. √ 1. The additive groups Z < Q < Q[ 2] < R < C.

2. The permutations of {1, 2, ··· , n} 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 ∈ H for all a, b ∈ H. T Cor 1.4. Let {Hi | i ∈ I} be a nonempty family of subgroups of a group G. Then i∈I Hi is a subgroup of G.

Def. Let G be a subgroup and X a subset of G. Let {Hi | i ∈ I} be the family of all subgroups T of G that contain X. Then i∈I Hi is called the subgroup of G generated by X and denoted hXi.

n1 n2 nt Remark. We have hXi = {a1 a2 ··· at | ai ∈ X, ni ∈ Z}. In particular, for a ∈ G, n hai = {a | n ∈ Z} 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 ∈ 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 = {a ∈ G | f(a) = eH }. It is a (normal) subgroup of G. • If A < G, then f(A) < H, where f(A) = {f(a) | a ∈ G} is the image of A under f.

• If B < H, then f −1(B) < G, where f −1(B) = {a ∈ G | f(a) ∈ B} is the inverse image of B under f.

Remark.

1. A homomorphism f is a monomorphism if and only if Ker f = {e}; f is an epimorphism if and only if f(G) = H.

2. For a ∈ G, the inverse image of f(a) ∈ H is

f −1(f(a)) = a · Ker f = {an | n ∈ Ker f}.