Groups
Chapter 1
Groups
The central goal of group theory is to classify groups up to isomorphism. This has be done for some restricted classes of groups, such as cyclic groups and finitely generated abelian groups, groups satisfying chain conditions and finite groups of small order. Groups, like some other algebraic structures, have important functions called homomorphisms, which play an important role in studying the structures of groups.
1.1 Semigroups, monoids and groups
A binary operation on a nonempty set G is a function G × G → G. It is usually written as a product: (a, b) 7→ ab. Sometimes it is written as a sum: (a, b) 7→ a + b if it is commutative.
Def. A semigroup is a nonempty set G together with a binary operation on G which is 1. associative: a(bc) = (ab)c for a, b, c ∈ G; a monoid is a semigroup G which contains a 2. identity element e ∈ G such that ae = ea = a for a ∈ G; a group is a monoid G such that 3. for every a ∈ G there exists an inverse element a−1 ∈ G such that a−1a = aa−1 = e. A semigroup is said to be abelian or commutative if it is 4. commutative: ab = ba for all a, b ∈ G.
In a group G, the identity e is unique, and the inverse a−1 for a given a ∈ G is unique.
Ex (semigroup). The set 2Z together with · is a semigroup. n×n Ex (monoid). Let Mn(C) = C be the set of all n × n matrices. Then Mn(C) together with matrix multiplication is a monoid.
Ex (group). The following are examples of groups:
9 10 CHAPTER 1. GROUPS
1. Each of Z, Q, R, C together with + is an abelian group.
2. The set Mn(C) together with + is an abelian group.
3. The set GLn(C) of all complex invertible n × n matrices with matrix multiplication is a non-abelian group.
∗ 4. The group of symmetries of a square, denoted D4, is a non-abelian group under the map composition operation.
5. The permutations of n letters {1, 2, ··· , n} form a group Sn under the map composition operation, called the symmetric group on n letters.
Prop 1.1. A semigroup G is a group if and only if both of the following hold:
1. For any a, b ∈ G, there exists x ∈ G such that ax = b;
2. for any a, b ∈ G, there exists x ∈ G such that xa = b.
The product G × H of two semigroups G and H is a semigroup under the operation:
(g1, h1)(g2, h2) := (g1g2, h1h2), for g1, g2 ∈ G, h1, h2 ∈ H.
This is the direct product of G and H. If the operation on G and H is +, then we write G ⊕ H instead of G × H.
Thm 1.2. Let ∼ be an equivalence relation on a monoid G such that a1 ∼ a2 and b1 ∼ b2 imply a1bl ∼ a2b2 for all ai, bi ∈ G. Then the set G/ ∼ of all equivalence classes of G under ∼ is a monoid under the binary operation defined by (¯a)(¯b) =a ¯¯b, where x¯ denotes the equivalence class of x ∈ G. If G is an [abelian] group, then so is G/ ∼.
An equivalent relation on G that satisfies the hypothesis of the theorem is called a con- gruence relation on G.
+ Ex. Fix m ∈ Z . Define a ∼ b in Z if a ≡ b modulo m.
1. (Z, ·) is a monoid. (Z/ ∼, ·) is also a monoid.
2. (Z, +) is a group. (Z/ ∼, +) is again a group, denoted Zm. In group theory, the congruence relation ∼ is often defined by modulo a normal subgroup of G. In ring theory, it is often defined by moduling an ideal. In module theory, it is often defined by moduling a submodule.