Quick viewing(Text Mode)

Groups

Groups

Chapter 1

Groups

The central goal of 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 . Groups, like some other algebraic structures, have important functions called , which play an important role in studying the structures of groups.

1.1 , and groups

A binary on a nonempty 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 is a nonempty set G together with a on G which is 1. associative: a(bc) = (ab)c for a, b, c ∈ G; a is a semigroup G which contains a 2. identity 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 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 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 .

2. The set Mn(C) together with + is an abelian group.

3. The set GLn(C) of all complex invertible n × n matrices with 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 of n letters {1, 2, ··· , n} form a group Sn under the map composition operation, called the 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)(, h2) := (g1g2, h1h2), for g1, g2 ∈ G, h1, h2 ∈ H.

This is the 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 , the congruence relation ∼ is often defined by modulo a normal of G. In theory, it is often defined by moduling an ideal. In theory, it is often defined by moduling a submodule.